Employment Law Questions? Ask an Employment Lawyer.
You can certainly attempt a filing, but the rule on being about to quit and get unemployment is that you have to have a situation so severe at work that any reasonable person would leave the employment. This is a sliding scale depending on the strength of your job market.
Now, it seems that you have a number of incidents to point to, but the state is also going to want to know that you did everything you could to stay, including confronting the perpetrator and essentially explaining that this sort of continued treatment would result in your having to quit.
Without that sort of evidence though (and your own statement is some evidence), you would have to make some sort of allegation of discrimination due to your race, religion, gender, age, disability or recent FMLA use.
The refusal to send the letter, while not illegal, is certainly another example of the behavior and the statements of other employees would be very supportive of your claim.
Then the issue will come down to whether or not the state believes it was reasonable to leave the employment under those circumstances. I won't lie to you here, these are difficult to win because the default is that resignation means disqualification. You bear the burden of proof to get the benefit, but you certain have the facts to support the argument and if you can get those statements you'll also have the evidence to support your position.