Your salary will enable you to live pretty well in any part of the city - alone or with girlfriend. If you're looking for trendy, go
Schanzenviertel or
Karolinenviertel, but be prepared to go to bed wearing earplugs. If you want a good mixture of proximity to shops and good ethnic mix, pick Altona (Bahrenfeld or Ottensen).
Speaking of salary, you should be able to find
a lot of info to compare with on the net.
Buying vs renting is a personal financial decision which I don't think anyone on a forum can make for you. That being said, what you get when you rent compared to what it costs to buy a place here puts renting clearly out in front unless you plan to stay here forever, and even then there's an argument to be made against buying. Again, it depends on your financial situation right now. If you do buy, be aware that mortgage terms here are very tight-ass. The banks here seem to be in collusion when it comes to the basic terms. Read the fine print because there is often a cap on the amount of principal you can repay in lump sums, which is a real rip-off longterm.
It can also be hard to find a decent place in the more sought-after locations. You often see home-made flyers up on boards and lampposts offering a reward of €500 or more for anyone finding the poster an apartment.
About your marital status: high taxes for single people, tax break for married couples, especially if one spouse doesn't have an income or is much lower than the other spouse's. It's called spousal splitting and works like this: instead of taxing your high 85K income at the higher rate, they split it into two. It's as if you and your spouse were each earning only 42.5K. Both your and your spouse's income are then taxed at a lower rate. Caveat: you have to be married. Living together? They don't care.