Is there a deadline to file for UI; Can I still file after a month?
No deadline, but you do not get paid retroactively. If you wait a month, you lose a month’s Unemployment Insurance benefits. Here is the answer if we change your question to waiting two weeks to file: The claim begins the Sunday of the week in which you file. So, if you wait two weeks, the claim will begin two weeks later, on the Sunday of the week in which you file for benefits. If you file on Wednesday, the claim begins 3 days prior on Sunday.

What will happen if I go on vacation for 2 weeks? Should I apply before or after vacation?
If you go on vacation once you have filed, a possible availability issue could arise. One of the criteria to receive benefits is; you must be able, available and actively seeking work. Are you seeking work while on vacation? Have you submitted resumes for employment during this time? In order to avoid this dilemma, an individual can file his/her claim after they have returned from vacation.

Are benefits paid weekly but certification takes place every 2 weeks?
Every two weeks you get a check for two weeks of benefits, after the first week, called the waiting week. You do not lose the money for the first week, as it is added to the end of the payments. If you signed up for direct deposit, you could expect the deposit in about 2 business days after you certify. And yes, you certify every two weeks.

What is the extension period for UI beyond 26 weeks due to COVID-19?
13 weeks, for a PEUC (Pandemic Emergency Unemployment Compensation) Claim. If the claimant is placed on an EB Extension (State Extended Benefits Claim), it is currently 13 weeks. 

If I receive furlough pay substantially below my regular pay, am I considered underemployed for UI eligibility?
Generally yes, however it would depend on what your weekly benefit amount is.

When will the extra $600/week COVID-19 UI supplemental pay end?
The supplemental payment is scheduled to expire on 7/25/20.

Who is eligible for the extra $600/week?
Anyone receiving at least $1 in UI benefits, regardless of the claim category in which it is received.