1) Are these nights really free? Do you have to pay a resort fee for these free nights?
The answer to every one of your questions is "it depends". On this one, depending on how much the casino likes you, you may get a discounted room, or a free room where you have to pay resort fees, or a free room with comped resort fees. Heck if they like you enough, you could also get free suites, free food / shows, comped airfare, or even a stack of cash when you walk in the door.
2) Can you get "free nights" by just signing up for the gambling loyalty program, but not gamble a single cent?
Generally no. If you could somehow convince a host that you are a mega whale (aka huge loser) at other properties, you might theoretically be able to pull this off, but I would say your odds are extremely thin.
3) If I practice advantage gambling on Blackjack, is it possible to get "free nights" and not lose any money? I read that with practiced card counting in Blackjack, the odds are in favor of the person gambling by 1-2%, so technically it's possible not to lose.
It depends on how you define the word "lose". Most people define this as leaving the table with less cash than they started with. Even the best AP can and will lose with regularity. This is a good thing if you are truly an Advantage Player. If you won every time, then the casino would stop letting you play.
If you define the word "lose" as "playing with a positive expected value", then yes it is possible not to lose. In any event, if you want to try the AP route, then you will need to invest money in some good books (PM me for suggestions), and also invest time in learning to play. You will also need a bankroll of at least $50,000 to play the games on the LV strip.
So will they still give comps if you are playing and winning?
Yes. The casino will give you comps as long as they believe that you are a losing player (playing a losing game) regardless of your short-term results. Some places will give you much better comps when you are losing. Since you have a group of 6 people, you can exploit this through opposition betting. One person bets banker and the other player (in Baccarat). You are hoping for a huge swing so one of you wins a ton (and gets no "bonus" comps), while the other loses a ton (and gets a lot of "bonus" comps). A bad result here would be a roughly equal mix of wins and losses for both players. Same strategy can be employed at Craps or Roulette. Please seek out a single zero Roulette table if you choose that game. By the way, act like you don't know your friend. The casino will not tolerate open team play in this manner.
Here are a couple suggestions. First, find ways to make it look like you are a bigger loser than you really are. Read "Comp City" by Max Rubin. The book is dated, but good. Make the pit think your average bet is higher than it really is. Slow the game down to a crawl (real easy to do at Baccarat), make blackjack plays that look really stupid but don't actually cost you a lot (hitting 12 v 6 falls in this category), use opposition betting to make your swings look bigger than they really are, etc.
Also note that you generally will not get great offers during your first Vegas visit. Once you have made a trip or two and convinced the casinos that you are a degenerate gambler with a huge trust fund, they will send you mail offers. If you are mainly interested in free rooms, this is what you are looking for.