Short version - yes, I believe people will consider it unprofessional.
Inability to properly transition
Historically, the primary reason for a two-week notice period is so that you can meet with a company-designated representative to explain your current status, how far you've gotten on your current tasks, what's left, etc. In some cases, you may be asked to show exactly how you do certain parts of your job if you've been the primary person responsible for them. Being out of the office impedes the company's ability to do that.
Future job prospects
You'd be making a very large change to your schedule in your final two weeks of work. This is one of the last things the company will remember about you. If this company survives and gets a phone call asking about you during a future job search, this may be something they mention and something the next company weighs into their decision. Alternately, the boss or your coworkers could end up at other companies that you apply to in the future, and may be asked about you.
Giving notice should reduce the immediacy of your issues
Finally, one would hope that giving your two weeks' notice would remove a large weight from your shoulders - of your listed issues, I would hope that #3 is gone altogether (no longer your problem), and that #5 and #7 wouldn't affect you as much. I would also like to think that most of the other issues would lessen at least slightly due to your being able to see the light at the end of the tunnel. As a result, I would suggest that you simply go in and put up with the remaining two weeks, knowing that it shouldn't affect you beyond that time.
I left my last job under somewhat similar circumstances - my coworker and I had been working several weekends, stayed late on several nights, and were heavily loaded. Once I gave my two-weeks' notice, however, I felt comfortable working regular hours again, leaving on time, and not working weekends unless I chose to do so. In the end, not all the work they had assigned to me got done, but I felt I could make a strong case that it was because they'd loaded us so heavily that we had no choice but to overwork to hit unrealistic deadlines, and the company couldn't say that I hadn't given a good-faith effort in my final two weeks.
Bottom line - it's probably best if you go ahead and go into the office, as tough as the next two weeks may be.
Caveat to the above
One major caveat is if things actually get worse as a result of your giving notice. If your coworkers' hazing turns into harassment, or your boss steps across the line in his treatment of you during your notice period, you should inform your boss (or HR, but you mention it's a small company) that you're willing to work through your notice only if you're treated properly throughout the remainder of your time with the company, giving exact references as to what happened and that you don't find it acceptable. If he cannot assure you that the problems will cease, then follow through and tell them that it's your last day. If you have to do this, write down exactly what happened, what you told your manager, and your manager's reaction. You'll want to be able to refer back to that in the future and you don't want to rely on your memory. Any notes about other times you attempted to talk to your boss about similar issues would also be helpful.
Best of luck to you, and good luck with the new position!