With some exceptions, I substantially agree with OzukakiBurasuki.
Quote:
If I lived in country side and I had enough land to bury them, I would have a grave in a yard,
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This being subjunctive, the wording is reasonably acceptable. Unless one prefers the academic "Were I to live in the country-side, with land enough to bury them in, they would have a grave in the (back) yard." that is. Note: it is the cats' graves, "I would have a grave" is unexpected - though understood. "I would give them graves" would pass without comment.
"When the cat died, I paid them 16,000 yen to them to cremate it." - "When the cat died, I paid 16 000 yen for them to cremate it." A variety of variants would be acceptable.
"but here in Kawasaki, people... " - I agree with the analysis for this sentence, but not with the proffered solution. "But" shouldn't be used to begin a sentence***, but "however" doesn't have that same restriction. " . However, here in Kawasaki" would preserve the contrastive structure.
still have a cat of my mother? - There were once three cats. "still have one of my mother's cats" - "mom" is American (and I believe, Canadian) English - it won't pass in British English or Australian without causing a raised eyebrow. "Mother" is perhaps formal, but it has international usage.
"So, of course the cat named Chihsuke is very cute." - "So, of course" declares the foregoing statement to be the reason for the stated following condition. Effectively, the statement is then, "the cat named Chihsuke is very cute because I still have her."
For ordinary purposes, the passage passes muster in its original form. Only careful reading will show any deficiencies.
*** Whoever made that rule ignored at least 400 years of normal usage in doing so. However, we are now stuck with the said ridiculous rule for written passages.