Beautiful enclosure, albeit total overkill for an avic (I keep adults in an 8x8x12...they would be perfectly fine in a vertical 2.5 gallon). Avics are known for webbing the very top of their enclosures, so it's likely that she won't use that hide on the bottom (not that it's bad to have, though...a few of my younger ones use terrestrial hides). This can make it tricky to access the tank if it has top-only access, but it looks like that particular top is hinged in the middle so you probably won't have any problems.
Avicularia are awesome beginner tarantulas. The worst you really have to worry about with them is their propensity for projectile defecation (enjoy cleaning that glass, lol). They thrive at room temp and higher humidity.
Do they try and escape a lot? I have the tank pretty well covered, but there are some really small gaps that concern me.
I wouldn't say they "try" to escape so much as they would if they're in the right place at the right time. I'd seal those holes.
As juveniles, every few weeks. As adults, probably every 6+ months. The time between molts increases with age.
The little info I found on the Web says that you need to look at the molt in order to sex these. Any input on that? Hopefully I lucked out and got a female.
When it molts, if you can, try and remove the exuvia as soon as possible -- the fresher, the better. Spread it out, get some tweezers, and gently unfold the scrunched up piece that was the opithosoma (abdomen). In between the first two book lungs, you're looking for a small "flap," called the spermathecae. If it's present, you have a female. If not, it's male. It's likely that you're going to have to do some research first to know what you're looking for, so do some googling. Once you get the hang of it, it's pretty easy.
Hope that helps!