No, the air doesn't come out. The caps are just to stop the valves getting crudded up.
Pressure warning lights are a bit temperamental in my experience, I've had them come on and pressures have been fine when I've checked them at home. I usually reset and only bother checking the tyres if they come on again. I keep an eye on my tyre pressures anyway, so it's not like I'm driving round with underinflated tyres.
Tyre pressures are also usually on a sticker under the fuel filler cap.
Second the recommendation to get a portable pump. My current and last car came with them - it was in the space where the non existent spare wheel should live. I'd check there before going out and buying one. Mine looks a bit cheap and crappy but is really good. I had a slow puncture once but it was a couple of days before someone could fix it for me so I had to pump the tyre up about 4 times and it did it in a couple of minutes.
I don't like this current trend for not supplying spare wheels, it seems a step too far in terms of cost cutting. If you have a spare wheel, you just pop it on and get on your way. Without, you can use the gunk that comes instead, but that ruins the tyre so instead of a chance of a puncture repair, you have to buy a new tyre.