Tire popping out of rim when inflating tube
Three things to check:

1, Is the bead seated correctly. It's pretty common to get tyres blowing off the rim if they're not seated correctly. That said, this usually leads to an exploding tube rather than the tyre just coming off
2, As Daniel R Hicks said, are the rims so worn they're expanding?
3, Check the rim for dings and warping as Malarky sort of says, but not just the area for the bead. You may have a bent rim

Chances are, though, that the tyres are done. The pictures would make me think they're on their last legs. I had a pair of 20+ year old ones recently that ended up in the bin as they just weren't reliable enough for me. Some things aren't worth saving a few euros/pounds/dollars over, and the contact points on a bike are some of those things for me. If the rims seem fine new tyres are where I'd go.