1. Yes. CS/Tear Gas and Pepper Punch would qualify as "a chemical", at least as far as I'm concerned.
The rounds listed are empty but can be filled with a chemical with a Logic + Armorer [Mental] (12, 1 minute) Extended Test for each round. A single dose of chemical fills 5 rounds and must be purchased separately.
2. Yes. All toxins would fulfill the "any compound" requirement of DMSO as far as I am concerned.
Any compound mixed with a dose of DMSO is deployable via the Contact vector (see p. 408, SR5).
3. As far as I can tell, that quote just means that a single dose of a chemical is sufficient for 5 rounds. I don't quite understand why they did this either as it implies less chemicals are needed for application by capsule rounds than is by disposable syringes or injection darts, but as there is no mention of a decrease in power it should be assumed, at least to my mind, that the stats of the chemical is unaltered.
4. Potentially, yes. By RAW, absolutely, as the rules for Capsule rounds on page 55 of Run & Gun are pretty explicit. Full Body Armor has the option of including a Chemical Seal, which would stop contact toxins from applying all-together, but otherwise it would seem that you are correct in your assessment.
Capsule rounds carry a liquid that, upon impact, splashes the target. Damage and AP are for the damage of the shot itself. A successful hit, including a Grazing Hit, means the capsule ruptures and the target faces damage from the chemical.
This one little rule makes capsule rounds, with the exception of the Light Pistol range they use regardless of which weapon they're fired from, superior in almost all ways, shapes, and forms, to other toxin application attacks including; coating a blade with a toxin, using injection arrows, bolts, or darts, slap patches, and disposable syringes. It's not clear from the text if the ammo is per 10 as in core or per round (though at 120¥ per round, EX-Explosive suddenly becomes a whole lot less attractive so it's somewhat fair to assume it's per 10), but the cost of Capsule rounds and their incredibly broad application makes them incredibly good choices.
Also, have some patience

You posted a question and didn't get an answer in less than 12 hours; sometimes things just take a little time...