If we are laid off but receive a job offer before the end of the notification period, do we still get severance?
That depends on you. You should negotiate your start date at the new job after the end of the notification period. If you can't do that, and you resign/quit your job at Cisco before your last day of employment, then you would forfeit your severance.
If we are laid off, the notification period ends, we receive severance, and then a few weeks later we get a new job offer—can we keep the severance?
Once you receive your severance, it's yours to keep. Nobody can pull back severance pay just because you found a new job after you are terminated.
How long do we have to wait before returning to Cisco without having to pay back the severance?
In most cases, you can't return to Cisco for 6 months, which is usually equal to the amount of pay in-lieu of notice and severance, so there's nothing to pay back. In some special cases, a VP? or maybe SVP? will want to bring you back before the 6 month black out period ends and they will negotiate whether or not you pay back a prorated portion of your severance or keep it, but this is a pretty rare situation.
Is severance paid before the end of the notification period? How can we be sure it will be paid?
Again, speaking from the past, normally you are notified and you are still employed, but not expected to work, for approx 30 days. At the end of that time, as documented in your HR paperwork, you will get paid however many days is left in what would have been a 60 days notice as "in-lieu of notice" pay. In the past, when we had PTO in the US, they paid out your unused accrued PTO balance, but that's now gone with the switch to unlimited VTO. They will pay back any deductions you've elected for ESPP purchases since you won't be eligible for the next ESPP purchase in June/Dec.
On your last day of employment, you are supposed to sign your agreement releasing Cisco from any lawsuits/arbitration claims, and in return for signing, you get 4 months of grossed up COBRA payments (regardless of whether or not you sign up for COBRA) and 4 months of severance. For those over 10 yrs of service, you get an additional week of severance for every year over 10, and 2 weeks for every year over 20. This is paid out within 2 weeks after receiving your signed agreement as you have 7 days to change your mind after sending it to Cisco.
How is the financial equivalent of unused PTO calculated when included with severance?
This is no longer applicable to the US. For other countries, your local laws would govern. In the past, it was based on how many hours you had accrued at the time your notice was given to you plus however many hours you normally accrue between the date of the notice and your termination date. Any scheduled time off that occurs after the notification date is canceled since you're not expected to work and are on paid leave until your termination date.