The first point is especially ridiculous because he looks and sounds obviously pissed off, obvious cope is obvious. I think I can safely assume the rabbit thing was faked too, we have no records of him even having one.
I'm torn on the rabbit. I actually thought the other pets stuff was BS from the beginning and either Phil being weird and talking about animals in his backyard, or maybe randomly bringing up his rabbit from when he was younger because he was desperate for sympathy.
The rabbit being sick is still super suspect to me because it happens out of the blue (unless DMs are missing, which is possible) and it happens in the shakier part of the leaks where Phil seems to have finally caught on that PPP wasn't his friend, so I think he's more likely to ramp up his lying during that period.
But very early on, during the "healthy" parts of their chats, where he really seems to treat PPP like a trusted friend, Phil mentions having 4 pets specifically, and before then, he said "pets" a few times instead of "pet". Obviously with Phil there's always the chance of him lying just because that's what he does, but it was an oddly specific moment during a very friendly period, which makes me think it's real. But he never once mentions the pets being sick, and given that he mentions Khet's health issues multiple times and says his mother has less than a year to live before he mentions the sick rabbit, it seems odd. Like, he gives the impression he didn't want to talk about the rabbit because it was personal, but...saying his wife/mom are dying isn't? There was also a point early on where Phil seemed sympathetic that PPP lost a pet, and Phil mentioned how he'd be so hurt if something like that happened to Jasper, but again didn't mention the sick bunny. Also, more recently, after PPP offered to fly Phil down to Florida, Phil does use his pets as an excuse for why he can't go, but he doesn't mention the pets being sick and says he'd have to get a pet sitter. I feel like if the rabbit was really as bad as Phil claimed, he'd have mentioned it during that part specifically.
Pets can get sick pretty suddenly and things can rapidly take a change for the worse, so it's not impossible there was a sick rabbit recently, but because of all the above mentioned stuff, I don't really buy it. This is also speculation because the rabbit details were oddly specific and Phil normally sucks at lies, but I wonder if he was using stories about "Bunny" to make it sound more legitimate. I don't know if there are any videos where Phil really talks about it, but maybe some Philstorian can see if the stories line up.
TLDR it's likely Phil has three other pets, but pretty questionable whether there was a sick rabbit