First off, let me tell you that no one will be able to truly answer this for you. As you've seen on the forums, the experiences vary wildly. here's my OPINION based on having this game since launch day +1 (and starting my first game with the 0-day patch).
1. Is the game entertaining and fun?
Hells. Yeah. This game has kept me up way too late the last 2 nights. Will tonight too, I bet, since I'll be firing it up after posting.

That said, many folks who game the system aren't feeling challenged enough. There are enough loopholes that you can bum rush or bling 'sploit your way to victory. I haven't done either, as I tend to turtle, research and steamroll on these types of games. And I'm loving it.
2. Will it let me develop strategies to win or lose against the AI? Today or Tonight?
I'd guess this has a lot to do with your experience level and how much you've studied the gaming system and your playstyle. I've only played on normal (I play epic length games when I do 4x), and though I've felt bottled in by random monsters I've not lost a city, not been attacked (despite being at war with 2 AIs for my whole game of many, many turns), and not felt at risk of losing. But this is normal, so I dunno.
3. Is the game polished and fixed enough to play decently?
Oh yeah, but only if you have patience for the occasional odd bugs. And the shards not powering up your magic really takes a lot of joy out of that part of the game (for me). But I still have 2 armies with 2-3 casters (kids, grandkids, 1 imbued champ) each that I enjoy using.
There was no 4.
5. Is the AI difficult and will I find challenges in against the AI in single player? Can I lose (not due to a glitch or bad luck... but because I was out played or out leveled or was a terrible diplomat)?
Not sure, I'd like to believe on higher difficulties you could. Honestly, if you know the system and game it, I suspect you'll be able to squash the AI on any level (based on the hardcore posters on this forum).
6. Replayability. Is there allot of replayability?
On my first game still, and am looking forward to replaying it, so I suppose there's some for me.
7. Bottom line: Is it worth buying it now - knowing about it being not superb and not very polished. Knowing that I will probably have 2 to 3 + crashes a night... is it still playable AND enjoyable enough to play?
I have 0 crashes a night, so that depends on your hardware/software. post 1.6, I don't even have slowdowns anymore. I do not regret buying this, and plan to play it on and off for a good long time. And I'll prolly be picking up Civ V on sale, not new, as they don't need my money as bad as SD does and I like to vote with my purchases. What you don't mention in this question, but do in your post, is your other RL concerns. So for me it makes sense, but for me the moeny isn't quite as tight as you made it sound for yourself. I'd weigh carefully if you'll get more joy from a happy mom and happy girl-of-interest than you would from a $50 game. How bad to you want to support SD? Will you play this in a few weeks after some more patches come out? I suggest you do, tho playing now is good fun too. If you chose to have fun with it, you will. If you get upset easily, or are one of the unlucky ones with compatibility issues, or expect too much from this small dev house, then maybe wait a bit.
We can't answer this for you, but hopefully something here has helped. I vote to go enjoy the girl if you care for her, but you don't need much money to do that (assuming she's a keeper kinda gal). But if you'll be playing anyway, there are far worse ways to part with $50. I know I'll be playing the heck outta this for a while.