- Yes. 2. Yes. 3. Maybe.
In general, you are allowed to deduct up to $50/month per student (see page 4), but only if you aren't reimbursed. In your case, since you are receiving a stipend, the full $2000 will be treated as taxable income. But the question of "is it worth it" really depends on how much you will actually spend (and also what you'll get from the experience). Suppose you actually spend $1000/month to host them, and if your combined tax rate is 35%, you'll pay $700 in additional taxes each month, but you'll still profit $300 each month.
If your primary motivation for hosting students is to make a profit, you could consider creating a business out of it. If you do that you will be able to deduct all of your legitimate business expenses which, in the above example, would be $1000/month. Keeping with that example, you would now pay taxes on $1000 instead of $2000, which would be $350, meaning your profit would now be $650/month. (Increasing your profit by $350/month.)
You will only need to keep spending records if you plan to go the business route.
My advice: assume you won't be going the business route, and then figure out what your break even point is based on your tax rate (Fed+state+FICA). The formula is:
Max you can spend per month without losing money = 2000 - (2000 * T)
e.g. if T = 35%, the break even point is $1300.
Side note: My family hosted 5 students in 5 years and it was always a fantastic experience. But it is also a very big commitment. Teenagers eat a lot, and they drive cars, and go on dates, and play sports, and need help with their homework (especially English papers), and they don't seem to like bed times or curfews. IMHO it's totally worth it, even without the stipend...