Peri:

Peri actvates his Eldritch Sight and follows Gerrard to the main deck. With the starboard side sitting higher than the port he figures they should start there (I don't see doors on the drawing so I'm imagining it's similar to the smugglers' ship, with 2 door forward and 2 more aft?) Nodding to the starboard after door and gripping his spear, he tells Gerrard, "Very well, Feldren. Open 'er up!"

Gerrard:

Nodding, Gerrard reaches for the door and carefully opens it, leading with his glowing blade.

There is a single door forward, and a single door aft. 

As you cross the main deck, Peri sees something arcane underneath with his Eldritch Sight, on a lower deck, along the port wall, roughly in the ships midsection.

Gerrard pushes open the door aft, under the quarterdeck, and peers into the dark cabin within.

A rough stone altar dominates this cabin. Strange markings cover its surface, dried mud cakes the floor around it, and humanoid skulls ring the wall above, each mounted on a wooden spike. The corners of this room are piled with garbage and debris — crudely cut wooden logs, smashed furniture, torn sacks, shattered crates, and small piles of dried palm fronds. A staircase descends from the southeast corner, and an acrid odor fills the air.

Gerrard:

Moving into the room, Gerrard looks carefully around for anything that could be the strongbox they seek.

Perception:  17

Peri and Kei follow Gerrard inside; Dwal and Oceanus stand just outside the door. Gerrard squints, scanning the chaotic mess around him, seeking the box.

It all happens at once: a massive spider bursts from the webs on the port side, while an upright insectoid emerges from the starboard side of the room. At the same time, four swarms of spiders boil up through the deck from below underneath three of you, crawling up your legs.

Everyone except Peri and Kei are surprised.

Initiative:

Kei:16
Gerrard: 14  (Surprised)
Peri: 11 
Dwal: 3 (Surprised)
Bjorn: 11 (Surprised)
Oceanus: 11 (Surprised)
Giant Spider: 8
Insectoid:  20
Swarms of Spiders:  11

 Peri may act first.

Peri:

Peri looks around the cabin in disgust. "Either this captain had some very exotic tastes or his is one of the skulls around the altar there. What kinda weirdo cult would make a--" His subconscious senses the shift in the debris an instant before the giant spider emerges, just giving him time to react. "Oh no! Nope! Everybody out!" he shouts as the spiders begin to boil up around their feet. Gerrard is staring confusedly at the spiders gathering near his feet, just sort of dancing in place, and hasn't noticed their mother looming up over his shoulder. Peri grabs the back of the noble's swordbelt, channels his Ki (Step of the Wind to Disengage), and yanks him backward through the door. "C'mon bard!" he yells at Kei on his way past. "No glory in gettin' ate by bugs!"

((Asking for a DM ruling: Peri uses his action to pull Gerrard free, and his BA to avoid Attacks of Opportunity. If Gerrard voluntarily moved, even if he was forced to move by a spell like Dissonant Whispers, he'd be subject to AoO; but if he was pushed out by a spell like Thunderwave he wouldn't. Do creatures still get their AoO on him if he's being dragged back? I'll pull him clear regardless because they'd all get surprise attacks anyway. And we'll end with Peri in the square below Dwal and Gerrard to Peri's right/forward.))

RAW:

You also don't provoke an opportunity attack when ...someone or something moves you without using your movement, action, or reaction.

Peri manages to kick his little (and big) attackers off, and pull Gerrard out backward, bowling into Dwal in the process:

Acrobatics checks:
Peri: 10
Gerrard: 10
Dwal: 17

 Dwal is fine (if maybe annoyed); Peri and Gerrard stumble and nearly fall prone, but manage to just keep from falling into a heap.

Unfortunately, while Peri is free, Gerrard is still covered in little spiders.

The two-legged insect thing leaps on Kei.

Bite attack: 22  Piercing damage: 6 plus 4 poison damage. (The target must succeed on a DC 11 Constitution saving throw or be poisoned for 1 minute. Kei may repeat the saving throw at the end of each of his turns, ending the effect on himself on a success.)

Claws Attack: 8 Slashing damage: 7

Kei may act next. (Roll CON save first!)

Kei 

((As instructed )) Con Save 22

((Can insect things hear? Do they even have ears?))

If they can: 

 If they can't:

 With the door blocked by a surly dwarf, he shouts as he tries to push his way out of the room. "Move it Dwal! I'm buggin' out of here!". He clambers over his short stocky companion. "CLOSE IT!" he yells as he ends up half hugging the busted mast, having made it across the Dwarven doorjamb.

After shaking off the toxins of a nasty bite, Kei whispers a horrid, mystically-laced insult to the hostile insectoid, hoping it indeed can hear them.
Wis save: 21

He then high-tails it out of the room, covered in biting spiders.
Opportunity attacks:
1. Insectoid: Bite Attack: 21  Piercing damage: 3 plus9 poison damage (plus another DC 11 Con save to avoid poisoning.)
2. Swarm Bite Attack: 9 Piercing damage: 11
 
Bowling into Dwal (again), he manages to shake off his attackers and make it to the mast. (Kei: Roll CON save again!)

Kei's little attackers find a new potential meal:  Dwal. They pounce.
Swarm Bite Attack (vs. Dwal): 11 Piercing damage: 8

The giant spider rushes to the door as well; it's appears almost too big to fit through, but it is extremely nimble. It snaps at Dwal as well:
Bite Attack (vs. Dwal): 13 Piercing damage: 4 (On a hit, Dwal must make a DC 11 Constitution saving throw, taking 13 poison damage on a failed save, or half as much damage on a successful one.)

The door is still open. Gerrard and Dwal are covered in swarms of biting spiders. The giant spider fills the doorway. The insectoid and the other two swarms are somewhere behind it. 

Peri may act next.

Peri is up. (Kei should roll Con save again for his second bite, then he will go next, followed by Dwal and Gerrard.)
If anyone decides to close the door, roll a contesting strength check.

Kei 

((In advance, but will wait to see Peri's action))

Con save 22

Peri:

Torn between stabbing the big mama spider, slamming the door in its face, or killing the droves of baby giant spiders, he figures Mama Spider is effectively plugging the doorway for now. So he decides to squish some baby giants first, starting with the ones clinging to their healer. (Spear: 14 for 8, followed by Spider Stomp TM: 12 for 6 + 11 for 7. If they die any additional attacks are against the ones on Gerrard.)

Peri manages to skewer or squash about half of the biting spiders crawling on Dwal. (The first two of his three attempts connected.)

Kei, who has again managed to avoid ill effects from the venom, may act next.

Kei 

Now clear of the bugs but having transferred them to Dwal, the bitter bard is quick to help his bearded friend. He focuses on all the bugs he can see, including the bipedal beetle. "Hey Dwal, make a wish. When you do, the bugs go squish". His poorly formed prose pushes Dwal's buttons to the point he could squish a bug just with his glare. (Dave, take a weird d6 B.I.)

Word of Radiance 10 Con 13 save. 

((Just reread this and it's confusing to me. So basically, Action - cast WoR. Bonus action - offer B.I. to Dwal.))

A burning radiance erupts from Kei as he utters his divine word:

Swarm 1 (Gerrard) - 14
Swarm 2 (Dwal) - 8
Giant Spider - 8
(with the giant spider filling the doorway, the bipedal insect and the other two swarms are no longer visible.)

Dwal is almost bug-free now, and the giant spider noticeably seared. But the swarms retaliate:
Swarm 1 vs. Gerrard: Bite Attack: 23 Piercing damage: 7
Swarm 2 vs. Dwal: Bite Attack: 7 Piercing damage: 6

The two other swarms arrive, boiling through the spiny legs of its giant comrade onto Oceanus and Peri:
Swarm 3 vs. Oceanus: Bite Attack: 20 Piercing damage: 9
Swarm 4 vs. Peri: Bite Attack: 21 Piercing damage: 9

Finally, the giant insect gets its hairy maw on Dwal again:
Bite Attack: 25 Piercing damage: 11 (on a hit, Dwal must make a DC 11 Constitution saving throw, taking 8 poison damage on a failed save, or half as much damage on a successful one.)

Oceanus is badly bitten on his neck and face. (Crit damage: 10). 

Dwal and Gerrard may act, finally.

Gerrard:

Finally recovering from the shock of the surprise attack, Gerrard is able to fend off the bites of the spiders crawling on him.  Calling out to Dwal, he says, "Let me take point, Brother Fiercehammer," and prepares to launch his counterattack.

OOC:  Gerrard goes after Dwal, so I will post his action in the spoiler below assuming Dwal moves out of the doorway.

Dwal

Dwal will step to the side (I guess over Oceanus' body), letting Gerrard attack.
Dwal will cast Word of Radiance on each creature.
CON Save: 13
Damage: 9

"The sea, now bugs!" he mumbles.

A burning white radiance flows out of Dwal upon his arcane command word.
CON save:
Swarm 1: 6
Swarm 2: 2
Swarm 3: 14
Swarm 4: 5
Giant Spider: 12

 The remaining spiders on Dwal shrivel up and drop away, and the ones one Gerrard and Oceanus are significantly lessened. Only the spiders crawling on Peri seem to have avoided damage, as did their giant leader.

Dwal steps away, between Kei and Oceanus (who, while significantly injured by his tiny attackers, is still upright and in the battle.)

The giant spider, however, doesn't let him slip away quite so easily:
Opportunity Attack: Bite Attack vs. Dwal: 15 Piercing damage: 7 (On hit make a DC 11 Constitution saving throw, taking 8 poison damage on a failed save, or half as much damage on a successful one.) 

Gerrard steps forward and sticks his Main Gauche deep into its midsection, seriously wounding it.

Oceanus does his best to ignore the biting pests still crawling on him, which have seriously torn up his face and neck. Instead he tried to plunge his trident into the big one with both hands:
Trident Attack vs giant spider: 7 Piercing damage:3 

Bjorn flies in and tries to poison the poisoner:
Sting Attack: 22  Piercing damage: 4

Bjorn's stinger connects.  DC11 CON Save: 7 

Peri is up next. (And Dwal should make a CON save.)