Can I collect rent while UD is in process?

If you take rent now the court may very well rule that you have waived your request for termination of the tenancy. It is always better not to accept rent after the notice period has run.

No. Once you accept rent you then are constructively renewing the lease (on a month to month basis) and would have to start the eviction process again either after another 30 day notice was given or if no rent paid, a 3 day notice to pay or quit is served. You do have a right to whatever rent the tenant should pay for the amount of time he holds over, but those are collectible in the court action.

It is unclear what your notice was for, but typically if you accept rent after the tenant unlawfully detains the property, that is treated as a waiver of the breach. If you have a lease that expires and the tenancy stays with your permission or consent, then you simply have a hold-over tenancy that is treated as a month-to-month situation. If the tenant was informed that you do not consent to holding over, then continuing to stay would be unlawfully detaining the premises. It sounds like you did this with the 30-day notice. If you accept rent, you would arguably be engaging in conduct showing consent for the tenant to stay, which would mean the tenant was staying with permission after all and therefore they would no longer be unlawfully detaining the property.

No. Collection of rent would be treated as your implicit cancellation of the 30 day notice, and would create a month to month rental agreement under the terms of the expired Lease. However, you can seek a judgment through the UD for rent owed through the date of the trial, if you requested that relief in the UD Complaint filed with the court. Typically, there is no notice required following the expiration of a fixed term lease, and no notice on non-renewal, unless this is a Section 8 tenant, or you are dealing with some special rent control laws.

Regretfully, No. If you collect rent after you started the unlawful detainer you cannot collect rent. You will be able to get a judgment for the rent until you receive possession, but if you accept the rent you will create a month-to-month agreement and waive the termination.

If you accept rent you must send him written notice that you reserve your right to proceed against him for any balance and possession.