As Earth was pulled down the gravity well of this dead star, our relative time would become increasingly faster compared to the rest of the universe. From our vantage point the time of the rest of the universe would slow down and all but stop, while we would experience millions of years of time passing relative to the rest of space. It would be like that episode of <3, Death, & Robots with the freezer civilization. We'd probably come up with FTL travel before our people could be turned into cosmic spaghetti, and be able to escape the Black Hole's event horizon.

Besides caves they also like to use hollow logs and the old nests of pileated woodpeckers. It's actually a problem these days in some areas that a lot of species that rely on dead wood can't find big old dead trees to live inside of because humans remove dead trees from their property. You can help out by putting up bat houses.

Northern yellow bats like to curl themselves up in Spanish moss on oak trees.

And my personal favorite are all the bats that make little tents for themselves by chewing on palm leaves to make the leaves sag into an A-frame. Like the Honduran white bat.

I'd say there is a not insignificant chance, if it is a brown dwarf. Though whether or not you would call a brown dwarf or a "hot Jupiter" a star is up for debate. In fact, two of the nearest brown dwarfs to Earth were only discovered very recently. Luhman 16 , a brown dwarf binary system, was discovered by the WISE mission in 2010. And WISE 0855 , a "sub brown dwarf", was found as recently as 2013.