Should I stay or should I go--on the verge of quitting

I am currently an analyst on an institutional group at a major BB. I am rather new to finance and was very excited to take the position when I was first presented with the opportunity. The team manages a large sum of assets (relatively speaking as compared to the rest of the wealth management community).

Unfortunately, the job has not panned out the way that I had previously hoped. The environment is toxic and I feel like it may be time to make my exit. I took the job roughly a year ago--I was previously working on the sales side at a mutual fund shop.

I have had my resume out and have been getting some pretty good traction. However, I am having to miss work to go on these multiple interviews. My job is not the type of position where missing work for any reason is really looked well upon. I also have a four month (yup, four freakin month) notice period that I am supposed to give when I do put in to leave this firm. It is very hard to interview while still keeping some sort of positive reputation at my current firm. I don't want to marked as a bad employee or written up for missing to many days, or because my work suffers because I am out.

My main questions are:

1. Since there is a four month notice period, should I put my "2 weeks" in now regardless of whether or not I have a firm offer yet from another firm?

2. Do firms usually make you stay for the entire notice period? I thought they would want you gone asap.

3. I feel like a job offer will come sooner rather than later, and I don't want my reputation tarnished as my current situation is quickly souring with my management--should I just exit asap and get my resume out as much as possible? I can't keep missing days to interview so I am in sort of a catch 22 .

Any advice would be very helpful. Thank you so much.