What are the limits of the Thin Man's poison spit?

The best way to think of the poison spit is as a grenade -- it can't miss, and is unaffected by cover. (And is also used when the Thin Men can't get a good shot due to targets being behind high cover, for instance)

The best way to avoid it is actually a bit counter-intuitive; Thin Men do not check if their targets are immune before spitting. I've had many a Medkit-wielding squaddie "tank" the thin men blast. Having a pair of soldiers adjacent is a poison spit magnet.

Since Thin Men are not selectable units in the Multiplayer mode, the easiest way to look into the mechanics of the ability would be to poke about in the modding kit.

First of all, most of your issues (barring the alleged infinite range) are equally applicable to grenades you throw, so it is "fair" for the xenos to have a similar weapon.

It's not that the poison spit has unlimited range, but rather that Thin Men have sniper-like perception and can generally see and shoot further than your units can.

Thin Men aim the poison at a unit (favoring those who have nearby friends). If you hide behind a wall (having no LOS to them), they could in principle aim next to your and have the splash get to you, but I have not seen that happen. So keeping your vulnerable (non-immune) units out of LOS while approaching helps.

Thin Men are my main priority in the early game. They are too able to sow chaos in my units for me to tolerate their presence any longer than I can avoid. When mid/late game more dangerous enemy types start appearing, Thin Men mostly stop appearing as often.

There are a few levels known to be heavy on Thin Man (e.g. Peter Van Doorn level). It's definitely okay to give everyone a medkit on those levels.

For the Peter Van Doorn level specifically, the spawn points are actually hardcoded. It's cheaty, but once you've done it a few times you know exactly where they will drop in, which gives you a huge edge by overwatching the right spots at the right time.

Always spread your units out, at least 3 tiles between them (of course, you have to balance this with the availability of acceptable cover) Immune units can of course ignore spreading out, but beware: if you group units together and not all of them are immune, you're actually making it more likely for your group to get poison spit and thus poison the non-immune unit.

You can use this to your advantage: group two immune units together. It's much more likely that the Thin Men will target your group instead of a single unit. And no, Thin Men do not know who is immune beforehand (nor do they remember it after trying).

When you're not expecting many Thin Men, it's often just better to not break your back trying to avoid the spit. Spread out and keep the usual amount of medkits you bring handy, but you're unlikely to get more poison spits than you get medkits.

If you do get more spits than once or twice, you're likely playing a very defensive game. Poison spit exists specifically to counter turtle tactics. Be more proactive and rout the enemy. It's better to get one unit splashed while attacking than to have the xenos chip away at your defensive turtling units.