I'll post a response in the hopes of reviving this thread and thus attempt to draw the attention of more experienced persons to provide their responses.
How do we get people to show up consistently to practice?
How do we improve the level at which the team can practice, while still having practice be challenging to varsity players?
If attendance is an issue, you might want to consider having two different types of practice a week. Have one where everyone attends, and practice answering easier questions. To avoid having your most experienced varsity players blitz JV players out of interest, rotate moderating duties - this gives the varsity players added experience of moderating, which will come in very helpful when the time comes to host and assist at other tournaments. (I'd also establish a gentlepersons' agreement amongst the varsity to not buzz before the question's done, but that might be too onery to carry out.) While you have one practice for all players, arrange a second practice principally for varsity. Encourage the JV to attend, but don't mandate it. The second practice would be pretty much what you have now amongst your varsity players. Having a more accessible practice geared for JV, co-run by varsity, could help encourage attendance.
Additionally, I'd say not making your practices too long. A former coach told me years ago that his practices wouldn't last more than 75 minutes. In my playing days (back when we used Kodachrome to take our state championship team photos) we had two, 2-hour practices. It was some good opportunities to bond with teammates, but in terms of time-effectiveness it was about the same as a narrowly focused practice of slightly shorter length.
How do we improve the skill of the team as a whole?
I was raised in the old school where list memorization was the main method of outside preparation, and we had quizzes at the beginning of each practice. You could do quizzes, or rapid-fire questions from the lists. One major caveat: with pyramidal questions, typically this only helps with the second-half of tossups. The best teams are going to have someone that knows a particular plot twist, historical quirk or chemical formula that's going to enable them to claim the potential power. You're going to need to encourage your peers to expand their horizons: read voraciously, take copious notes during class, pay attention to the credits at the beginning of TV shows (yes I did power a tossup that way).
How do we get people to go to Quiz Bowl tournaments over Speech and Debate tournaments?
If memory serves me correctly, attending quiz bowl tournaments should also count for NFL participating points. I'd check with pertinent sponsors on that. That may prove to be an attractive option for players who aren't getting as much competition time at speech/debate/forensic tournaments.
Do you have any advice for convincing people to sign up for tournaments?
You should be able to convince your activities director that students participating in quiz bowl qualify to receive a letter (if you aren't already - if you have a stubborn AD that won't consider it despite letters being available for orchestra, debate, etc., go to the next level!) Having your advisor establish guidelines to earn a letter could provide an additional incentive for your peers to participate.
An entry-level or open practice would help get people interested and eventually get the training wheels off.
There are several other ideas worth floating, so I'll leave this as an opening suggestion. Other thoughts, folks?