IMO that sounds like way too much for that sized apartment (even without the increase). If you can stomach the headache of moving, that might be your best bet. I'm sure you'll more than make up for the moving costs even within a few months of finding somewhere cheaper.

If you are not certain moving is the best option for you, you could try to negotiate with the building management/landlord.

As for your other questions; 2. Double check the language in your original lease, but my bet is that you'd have up until at least a month before your lease expires before you'd have to resign. Most likely you could stay through the end of your current lease before signing anything new (though that could get stressful if you end up deciding to move). I also don't think they'll come up on the price increase they've already given you. Giving you that 3/1 deadline is a common tactic to make you feel like you're actually getting a deal (when you're not) but you need to act fast or it will disappear. In reality, if you were to ignore them until after 3/1, or just outright say no, they'll likely come down on the price increase.

Question 3. As I alluded above, I think you're quite likely to find either a studio for less than you're paying now, or a bigger place for a similar price. If you want to stay in the neighborhood, moving a block or two from the park could potentially reap you some significant savings.

They want you to panic and make an impulsive decision- try not to do that.

Best of luck!