If my husband had the mumps twice as a child, is it likely that his testicles are sterile?

Depends on when ... Getting mumps after puberty is more likely to affect his fertility - even then, not all boys/men have infertility or reduced fertility. He needs to have a semen analysis, preferably through a fertility clinic. 30 to 50% of males with mumps affecting the testicles get testicular atrophy, where one or both testicles reduce in size. The chance of sperm issues is at least 15%.
RARELY IF EVE. There was a time when mumps was common in childhood and those kids grew up and rarely had fertility issues. For someone to have it twice, I would suspect at least one episode was mis-labeled.Salivary gland infections do recur, but mumps rarely if ever would.Teen or adult mumps is am uncommon bur recognized cause of male infertility.A simple sperm study can confirm or exclude this as a factor.
Not necessarily. Obtain a semen analysis and you will know the answer. I recommend obtaining it in a facility that can freeze sperm too so that if the numbers are small they can freeze the sample right away.
Possibly... ...But not very likely. Adult mumps is more significant as a cause of sterility.