I have two questions question about my first session as DM. I played before, but my previous DM never used maps (or we somehow never arrived at this point):

1)

[ o ] [ o ] [ o ] [ o ]

[ o ] [ E ] [ o ] [ o ]

[ o ] [ X ] [ A ] [ o ]

o = empty, E = Enemy, A = Ally, X = player. Every [ ] is 5x5 feet.

So a player wants to flank an enemy together with his Ally A. Can he walk around E to the left upper corner and does this trigger AoO's? (All have range of 5 feet.)

I ruled the first instance in the players favor, even though it felt weird that while allready attacking, a player could disengage, move and attack in the same turn.

2)

[ o ] [ E ] [ A ] [ X ]

[ o ] [ E ] [ A ] [ o ]

Same question, now in a narrow hallway. If the allies "make space" and the player is a small humanoid, can she somhow pass the allies and enemies to the left upper corner?

I had no idea what happens in the second one. I used the 'rule of cool', asked the player how she wanted to do this. She described sliding through between legs of players of the allies and opponents, rolled a high acrobatics check so I allowed it.

Thanks upfront! Also, if someone can point me to a guide how to upload images to this forum, that would be nice! 

Lyxen answered you - but just for completeness - here's some quotes from the rules.
For 1):

https://www.dndbeyond.com/sources/basic-rules/combat#OpportunityAttacks
Opportunity Attacks

[...] You can make an opportunity attack when a hostile creature that you can see moves out of your reach.

If you don't leave a creature's reach - you don't provoke an AoO.

For 2):

https://www.dndbeyond.com/sources/basic-rules/combat#MovingAroundOtherCreatures
Moving Around Other Creatures
You can move through a nonhostile creature's space. In contrast, you can move through a hostile creature's space only if the creature is at least two sizes larger or smaller than you. Remember that another creature's space is difficult terrain for you.

So the Small character can only move through a creature who is Large or larger.

Okay. My previous DM played 3.5 and introduced this rule. I could not find it in the rules, so everyone was confused but we never asked about it.

So. Follow-up question, how does a rogue forces a sneak attack? I know that a sneak attack is allowed whenever the rogue has advantage. But how does it gain advantage, if not for flanking?

It doesn't need advantage.

https://www.dndbeyond.com/classes/rogue#SneakAttack-343
Sneak Attack
[...] You don’t need advantage on the attack roll if another enemy of the target is within 5 feet of it, that enemy isn’t incapacitated, and you don’t have disadvantage on the attack roll.

So you have 3 situations:

1. You have disadvantage on the attack: Cannot Sneak Attack.
2. You have advantage on the attack: Can Sneak Attack.
3. You have neither: Requires an enemy of the target to be within 5 ft of the target in order to Sneak Attack. (Note: It doesn't say ally of the Rogue - it says enemy of target - the distinction can sometimes be important.)

Of course there are always exceptions to the rule. There are a couple of Rogue subclasses (like the Swashbuckler) that add their own extra specific cases where they can Sneak Attack besides the above rules.

Quote from CastlesAndChimeras >>

Okay. My previous DM played 3.5 and introduced this rule. I could not find it in the rules, so everyone was confused but we never asked about it.

So. Follow-up question, how does a rogue forces a sneak attack? I know that a sneak attack is allowed whenever the rogue has advantage. But how does it gain advantage, if not for flanking?

The one way a Rogue would use most often is attacking while unseen. Rogues can use a Bonus Action to Hide and, if successful, they are then unseen. This tactic is better used for ranged attacks, because coming out of a hiding spot to approach for melee usually puts you into direct view of your target and thus negates your hiding.

Other methods of gaining Advantage: Attacking a prone target from 5ft away, benefiting from the Help action from an ally, attacking a target under the effects of Faerie Fire, using the True Strike (don't do this, it's usually not worth it), attacking a barbarian that has used Reckless Attack.

Also note that rogues can benefit from Sneak Attack when their target is within 5ft of another creature that is hostile to the target, even if they don't have Advantage. So teaming up to attack a target that an ally is also attacking is still beneficial, even if it doesn't get you Advantage. This also work for ranged attacks. You do not need to be within 5ft, you just have to have an ally within 5ft of the target.

There are variant rules in the DMG that DM's can choose to use, or not.  One of these variant rules is Tumbling:

https://www.dndbeyond.com/sources/dmg/dungeon-masters-workshop#ActionOptions

Tumble

A creature can try to tumble through a hostile creature’s space, ducking and weaving past the opponent. As an action or a bonus action, the tumbler makes a Dexterity (Acrobatics) check contested by the hostile creature’s Dexterity (Acrobatics) check. If the tumbler wins the contest, it can move through the hostile creature’s space once this turn.

Guys, you are all amazing. Truely helpful! Can I persuade you to be stand-by during our next session? ;)