Not in the sense people usually use the phrase "action points".

In D&D, on your turn, you get one action. That might be hitting someone with a sword, casting a spell, or whatever. You can also move, but that's not typically an "action". Some classes also get what are confusingly called "bonus actions", which are extra things you can do on your turn. Rogues, for example, can try to hide as a bonus action even after they did a normal action like shoot someone. A regular fighter who shot someone with a bow wouldn't be allowed to try to hide on the same turn, but the rogue could.

Because you only get one action (and bonus actions), no one calls them action points. You might say "you already took your action" or something.

Compare with a game like the original Fallout. You get a number of action points based on your stats. For an average character, it's 7. Shooting someone typically takes 2, a called shot takes 3. I think movement also takes action points. So on your turn you might shoot someone 3 times (six action points) and move a little (1 action point). Or take a called shot (3 AP) and move a lot (4 AP). Lots of options. Divinity Original Sin 1 and 2 also use action points in a similar way.

This is way more explanation than you probably needed. The key thing is on D&D you typically do one thing on your turn (eg: attack, cast a spell). Pretty much every action has the same "cost" in this sense. You can't really mix and match, and there's not as much tactical depth on your turn. (Pathfinder 2e gives you three actions per turn, but I don't think anyone calls them points there, either.)

Yes but not in that way.

Initiative determines the order of action. Usually in combat, but it could theoretically be applied elsewhere, a debate for example, or if a Bard challenges someone to a rap-battle (I guess those are both kinda just verbal combat though).

Action points are how many actions you can perform per turn of initiative. Usually a movement, an action, and possibly a bonus action.

Slipping on a banana peel could be done if you rolled acrobatics to leap over the banana peel but failed the skill check. Say you're a Gnome whose lost their legs, taking a -6 modifier to Dexterity. This could cause an acrobatics roll of 12 to fall to 6, which could below the 7 you might contextually need to succeed at leaving over the banana peel.

"Using your powerful gnome arms you make a mad gorilla dash and leap with all your might, Unfortunately the loose dirt under your right hand gives way and you slip. You manage to catch yourself with your left hand, but before the squelch of the banana peel registers to your ear your arm slips out from under you and you strike your head upon the ground"
Take 1 Damage and Roll a Con save to see if you're concussed.

Edit: I brainfarted and described a potential DnD or PF situation, this cannot occur in BG