"According to the calculations all the gain on new home is excluded."

Where did you figure that calculation?

"Working through the calculations, it appears that the original $500,000 exclusion was reduced by gain on 1st home, and then 2nd home gain was not enough to use up all the exclusion"

Where do you see that is a carryover limit?

"Lived in 2nd home 16 months."

So, does not qualify for primary residence living there for 2 out of 5 years of ownership.

"due to unforseen circumstances."

Have you confirmed with the IRS list if that qualifies for the exception from taxable gain, at least partially?

"Assuming this is correct, is the basis on the new house (3rd one) reduced by both the gain on 1st house and second house?"

Are you reading something from 30 years ago, or current materials? Try these:

https://www.irs.gov/newsroom/important-tax-reminders-for-people-selling-a-home

https://www.irs.gov/businesses/small-businesses-self-employed/sale-of-residence-real-estate-tax-tips

https://www.irs.gov/taxtopics/tc701

https://www.irs.gov/publications/p523

(the pub covers Unforeseeable Events)