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In terms of unemployment, you can file, but you would have to report the income from all of your part time jobs for each week that you receive it, which will very likely result in you not receiving any unemployment. The state doesn't actually consider how much money you lose when deciding what your rate of unemployment is. If your income from the other sources is too great, you'll get no benefit from unemployment. However, if you voluntarily stop working those jobs, you'll essentially disqualify yourself from unemployment too.
You would need to be filing for unemployment the entire time, even if you aren't paid anything, just to keep the benefit going. The only way you can move into the longer periods of unemployment is to have filed, during the appropriate time frame, for the earlier benefit (even if you're income from the part time jobs is too high for you to actually be paid anything).
On the issue of severance, unless you have a contract of employment that specifically stated that you could only be terminated for cause, the employer doesn't legally need a cause for termination in your state. This is called "at will" employment, which is the predominant employment status in this country. Just being terminated without cause is not grounds for demanding severance. You'd need some indication that the basis for your termination was race, religion, gender, age, disability or FMLA use.