Some weeks ago, ...
Obviously, this happened in the past, and we don't know if the husband is still alive or not. We're assuming he is, but to my mind, both present and past would work here. Initially, I had the present tense here but changed it.
She thinks about eternity etc. - what's so odd about that? We can safely assume that Stephen Hawking, for instance, thinks a lot about the universe. We don't know if he's actually thinking about it this very moment, though.
The garden is mentioned in the very first sentence, so we know which garden this is. I see no problem to then refer to it as "that garden." If you prefer "the" garden, that's your prerogative, but "that" isn't wrong.
"Sich kümmern" tends to imply more than just sitting around the house, watching TV. Whether the son is a professional nurse or simply a son who cares doesn't matter so much. To the mother at least, he takes care of her husband, so I would assume it was important to her to mention the "sich kümmern" part, important enough, perhaps, for the writer of the letter to mention this as well.