In 2e, traps are there specifically to slow down the party and force them to be cautious. Even in 5e, traps are a break in the action and can be triggered before anyone sees them. The 2e style doesn't sound like your playstyle at all, but even the way 5e uses traps by default seems to not be your style either. So...
Tell them the trap is there
Traps are interesting when players interact with them. Tell them the trap is there! Yes, 5e has Passive Perception for this purpose, but don't let that ruin your fun if giving the check away "for free" will significantly improve your and your players' enjoyment of the game. (Passive Perception is still useful for surprise encounters and ambushes, after all.)
By telling them that a trap is present without telling them exactly where and what, you get straight to the interesting interaction, as they investigate the trap to figure out where it is, what it does, and how to circumvent or disable it. They may play it smart or they may make mistakes and suffer the trap's consequences. Some traps will be boring (and quickly dispatched) and some will be super-interesting (and be super-interesting). Either way—smart or disastrous, trivial or super-interesting—you are keeping the part of traps you find fun (the players messing around with them), eliminating the part you find frustrating (caution bringing the dungeoncrawl to a literal crawl), and quickly moving along to wherever the fun is actually to be found.
So skip straight to the fun, and tell them that there is a trap here. Let them figure out what kind and where exactly, but tell them:
As you move to cross the threshold, your keen adventurer's survival instinct tells you something is wrong. There's something not quite right about either the door's stone frame, or the room's floor… or maybe it's something else nearby that's off. Anyway, what do you do?
...and cue the flurry of investigation!
DCs for the traps, should then be converted like most DCs in 5e. 10 for easy, 15 for moderate, 20 for hard, 25 for very hard.
Perception is for observation, Investigation is for deduction.
Some of this answer will be observations on how Wizards has done it so far and some of this will be logic, and some of it will be mechanics.
First for the headline question. It depends. when to use Investigation, and when to use Perception is not entirely clear yet, and I'm hoping we'll get more guidance in the DMG late this year. The guidance from the rules is that the two skills mostly seem to differ in the methods by which they are found.
Depending on the exercise, either, or both of the skills may be used.
For passive checks, you're almost always looking at Perception being the skill of record. While all skills can be used passively, some skills make more sense than others. Perception is the hallmark passive skill, whereas investigation makes less sense as a passive skill.
If the character is alert to the possibility of hidden objects/traps, but not actively searching, he's using passive perception.
The guidance for this seems to be (though we can't confirm yet), that the DC for actively looking for something is regularly about 5 less than the DC for passively looking for something (or sometimes actively looking always reveals it).
So here's how I would play it. Traps can be noticed with perception passively (usually DC 15). If the PC is looking, ask them how they are looking. If it's observational, then use Perception. If it's deductive, use Investigation. When they are searching for items, again, either skill is appropriate. This is somewhat counter to how WOTC has written adventures so far. They are always written to use perception to notice traps. Passive with a higher DC and active with a lower.
Investigation also has broader uses such as when you are trying to track clues, or put something together. It's also a great "roll for a hint" kind of skill if your PCs get stuck and need some help figuring out what to do next.
This is covered in the Basic Rules, Chapter 8 (Adventuring) under Travel Pace (my emphasis):