Does the Spider Climb spell still work on a wizard that wears a full-body gorilla suit that covers the hands and feet?

I don't see why not

So, any humanoid spellcaster casting this on themselves could just climb up a wall, and keep casting other spells from there for an hour? 

Yup :) It is very likely so.

Why doesn't this increase number of targets with upcasting, like Invisibility and Fly do?

If I take this as a Paladin with heavy armor, do I still have to make athletics checks to climb, or can I just "walk up the wall now", like the rogue and other classes, that don't carry as much weight, would?

Can someone with spider climb end their turn on a wall or upside down?

Quote from SockoFaye13 >>

Can someone with spider climb end their turn on a wall or upside down?

 A good question, the wording is ambiguous.

Quote from Qoray >>

If I take this as a Paladin with heavy armor, do I still have to make athletics checks to climb, or can I just "walk up the wall now", like the rogue and other classes, that don't carry as much weight, would?

 It’s not a Paladin spell, but if someone cast it on you:  yes.  Barring extraordinary circumstances, you can just walk on walls, with no speed reduction.

It’s a nice little spell.

Spider Climb + Simic Hyrid with Manta Glide :D

Plus maybe Thorn Whip with Spellsniper for an extra dose of trolling...

I thought spider climb worked for rouges

You can take it as an arcane trickster, but only as one of your "spells from any school." You can, therefore, choose spider climb at rogue level 8, 14, or 20.

You can also replace your rogue level 3 "spell from any school" with spider climb once you have access to second level spells at level 7.

Does this work for Centaur players?  Asking for reasons.... 

Yep, basically with this spell working, you’re not climbing at all, so the penalty from centaur doesn’t apply. You’re just walking up walls, ceilings, etc with no more effort than it takes to walk on level ground.

Edit: not quite right... it does say you gain a climbing speed ... so now I’m confused lol

Spider climb requires concentration, so you can’t cast any other spells

You can cast other spells as long as they don't also require concentration. And remember, if you are hit, you'll need to have a Constitution save in order to retain concentration, which could see you falling from great heights. 

I'm doing a cleric/moon druid, how can I get this spell with out multiclassing again?

Spider Climb + Mending = great on a ship. Prancy around the outside of the hull, and mend any scrapes and fissures as you go. Quite meditative really.

You cant, but one thing you want to look at is the slippers of spider climbing. 

Quote from Eravanfeighn >>

Spider climb requires concentration, so you can’t cast any other spells

 You can't cast other concentration spells, but sure you can cast tons of other spells, as long as they don't require concentration.