It sounds like you'll be staying north of the Golden Horn (the waterway that runs east from the Bosphorus, the Bosphorus being the waterway that seperates Europe from Asia). The area at the north end of Istiklal Caddesi is Taksim Square. Taksim and Istiklal is my favourite place to stay, especially as a single woman. The streets are quite busy in the evenings, allowing you to safely leave the hotel to grab some food or run to a convenience store. This was not true of Sultanahmet, where most of the tourist sites are. Sultanahmet may be safe, but it's deserted after dark.
Dolmabache Palace is just north of this, and I highly recommend visiting it. When the Ottomans built it, they abandoned Topkapi and brought all their stuff with them. As a result, Topkapi is rather spartan. Topkapi is also the most expensive attraction, so it was a big disappointment for me.
The taxi from Taksim to Sultanahmet was about $8 last January. Read all the advice on here about taxis, but they're still really cheap and the easiest way to get around.
I didn't encounter any issues with sexism, in terms of men refusing to serve me, etc. I also didn't encounter any harassment, but it was January, so I wasn't walking around in shorts and tank tops. Salesmen are much more aggressive than you would be used to in the US, and some people consider it harassment. It's also possible that women are targetted more, but they're really more interested in making a sale than anything else.