Our gut bacteria is vital so are we eating so they can process it for us or are we eating to feed them?

Commensalism :)

https://en.wikipedia.org/wiki/Commensalism

Lots of these bacteria probably exist without us. We can’t exist without them but we do make a very nice house where they can live.

While some bacteria definitely can, there are others that are so adapted to building colonies inside of us that they can’t really survive outside of a mammal or human host. You can carefully create the lab environments to allow for it, but that is against the spirit of the question. My gut says ( ;) ) that anaerobic bacteria that evolved for mucosal colonization are likely to fit into this group.

Meanwhile it is possible to survive without much of a microbiome, but you end up having trouble with digestion and you are susceptible to infections since the surfaces don’t have any residents and are “available”. Newborns have low microbiome diversity and build it up rapidly after birth, and people on sustained antibiotic treatments tend to lose a lot of their microbiome.

So they keep us around.

Here is another fun one, the symbiogenesis theory.

The mitochondria was likely an independent prokaryote (bacteria) that was taken into eukaryotic (plants / animals / etc) cells.

https://en.wikipedia.org/wiki/Symbiogenesis

Even now, the mitochondria has its own genetic material and replicates somewhat independently. So if you went in and removed all of the mitochondria from a cell, it wouldn’t be able to make a new one. That is also why your mitochondria will match the mitochondria from your mother, since the egg cells come with mitochondria while the sperm cells don’t.