Yes, you have to pay your own way. No, you don't have to pay them $1000 a month. I don't know where the hell that idea came from, unless you plan on spending that much on the toys and trinkets you buy on your off time.
You live on base, you eat in the standard mess, and you get 1-2 days off a week to travel around. While the web site specifically talks about hospital work, most of those I know who've gone ended up at an IDF base doing maintainance and repair type stuff.
No, you CANNOT bring a firearm into the country. Only Israeli citizens are allowed to carry, and the police have to approve all private firarms transfers. Yes, folks on the West Bank are often armed, but you won't see many (if any) armed civilians on the streets of Tel Aviv, Haifa or Jerusalem. The only armed civilians I know of in those areas are security officers. Think of Israel's firearms laws as being more like New York or Massachusetts, than Georgia or Texas.
Re Americans serving in the IDF, only if they have dual US/Israeli citizenship. That's something fairly new in US law (policy?). Back when I was there for an extended period in 1956, you could get threatened with US citizenship revocation if you even carried a rifle on guard duty at a kibbutz-----which is why I never discussed my experiences there with anyone, even my family, until just a few years ago. Don't know when the dual citizenship thing went into effect here. Israel (and many other countries around the world) has always had it.