In general, things pretty much take care of themselves, I think the worst part is dealing with the the Hewitt YBR who manages the COBRA and benefits, their customer service is very bad and limited.
Company Phone - You can try to request the COP cell number to be handed over to you, probably too late, I attempted to do so, but IT cuts that off several months ahead of any layoffs. Worth a try if you wanted to keep that number; however, getting a new phone setup prior to layoff is helpful if the COP cell was the only phone you had.
COBRA - I was on COBRA medical benefits until I found another job. Setting it up was easy, however, cancelling can catch you. They set it up so that you pay for the insurance at the start of the month and you can not cancel ahead of time, you will need to call the day of that you want the insurance to end. It is a lengthy process to get your money back when you cancel. If I had to do it again, I would skip the autopay and have them bill me to stay in control of the cash flow. By law you have several days after they bill you to pay COBRA, and if you dont pay after a specified time, they can cancel it.
HSA - You can keep the HSA with COP but they will start charging fees. I moved my HSA to ModernWoodman Bank which thus far has been better than the COP HSA, no fees and they are easy to work with. COP does not fund HSA as part of the severance or final pay check. If you wanted to max out the pre-tax contribution to HSA, easiest way is to max out while still at COP and be done with it for the year. You can make contribute after the fact, which you have to do with your HSA and account for it in your tax returns.
401K - left it with Vangaurd, had no issues since, no extra fees or different customer service since layoff. You can roll this into another 401K or rollover account if you choose. The only small downside with a rollover into a general IRA is that you no longer have loan options against your 401K if you wanted that feature. If you intended to put in the max contribution to the COP 401k, then suggest you do that ahead of layoff. I think an post layoff 401k contribution becomes tricky with taxes and other rules.
RSU/Stocks - I had stock vested with my severance, I left this with ML and as with Vanguard, no extra fees or problems with customer service. You can move this to another broker if you wanted.
Retirement/Pension - You can keep your pension with COP or cash out and put a lump sum into your 401k or other account tax free.