I'll answer in parts...
From what I've read, the IMR batteries these use typically output 3.7 volts.
Not just IMR. Any Lithium-Ion battery (standard cells) have an output which is 3.7v typical. It ranges from 3.3v when discharged and it goes to 4.2v when fully charged.
That's just enough on my vv battery to be satisfying, but I'm wondering if it works differently on a mech mod.
Yes, is very different. Any VV or VW device accepts a range of voltage input, and you can control the output.
For example, on a VV battery you can set it at 4.2v and the output will always be 4.2v regardless of the internal battery voltage.
On a Mechanical mod you get directly the battery cell voltage. So the vape experience will be beter or worse depending on the charge.
I've heard terms like voltage drop or something, and I'm not sure what that means.
At first you shouldn't worry about this. It happens on every mod, atty, and every contact. It doesn't affect only to mech mods.
If you don't want more than 20W, the voltage drop shouldn't be an issue for you.
For more, you should get mech mods made of cooper with good contacts, you can ask to the experts here

Anyway, it's a small problem; you don't have to worry about it.
I also don't know too much about kicks. I hear that they're another mechanical device that can up the voltage of your IMR battery. But I'm not even sure if the mod I want has room for a kick. Do all mech mods have room for kicks or do you need like a special telescoping one to fit it properly?
Kicks are electronic devices. So when you put a kick in the mech mod, you don't have a fully mechanical mod.
They are a piece of circuit that does what your VV device does, but in a really small space. They are convenient for beginners on mech mods, because they allow a fixed Wattage (more consistent vaping) and some protections.
If you want to use a kick, it will fit or not depending on the mod and the battery size you install on it. The most problematic for kicks is the 18650, because it is the larger one and they usually target mods for exactly its size. Telescopic mods have more probability to work with kicks, but ask/search anyway for each model you're interested. If you use a 18500 or 18350, any telescopic that fits a 18650 should fit your batt + kick.
I have a V2 Lanucher + VW kick, using a 18350. It works good, and protects from reverse battery and protects from shorts. My girlfriend is happy with this combination, and she vapes it using it at max setting (15W). The downside is more contacts and from time to time the mod doesn't fire because the kick doesn't have contact to the battery or to the mod. Nothing to worry about.
If you plan to use atomizers with resistance lower than 1.5ohm or you want more than 15W, then a kick is not for you. It's hard to find kicks that work with higher wattages or less resistance.