Issue 1: Will people complain that tryouts are rigged or otherwise unfair if she makes the team without being able to compete? She is going to attend to help out, but her doctor said she's a no go for participating in any of the skills evaluations. It's also not fair to her to keep her off the team if she's going to return shortly after tryouts.
This isn't an issue. You can't care what others think or more importantly you can't take what other think into consideration for a decision that is yours to make. You have already said that #15 is going to be someone who hasn't played before - keeping someone who CAN play - and can play an important position - is a no-brainer no matter how her injury status works out.
Issue 2: I have no idea how long she's out. If she's only going to be out a couple weeks, I'll take 14 plus her and we'll just play a girl short until she's back.
This is also not an issue. Your cuts are not travel or even rec players by the sound of it. People probably have no idea what the # of players you are allowed to keep is. You may not even keep 14 of the tryout girls if there are multiple 'never played befores' in that 14.
Do I talk to that last girl cut and offer her the chance to practice with the team the first week, knowing her spot on the team is contingent on another girl's health (this seems like a really bad idea!)?
I would not do this. This girl isn't going to make or break your team this year or in any future year - especially when you seem unsure that some of the players who HAVE played before are good enough to beat out a player who has never played before.
My one concern in this sort of situation is that 20 @ tryout can dwindle REAL fast - you then select 14 and 4 of them decide it isn't worth it.... now you have 10... Be wary of the 'playing/trying out because my friend if playing/trying out' dynamic. At tryouts you may have to separate the 'played before' from the 'never played' and run basically 2 tryouts - so you may need help there.