The dogs do die, but there's an option to disable dog death entirely or extend the dog lifespan in the settings menu. If you don't want to be heartbroken, turn dog death off or set the lifespan really high, but dog death disrupts the simulation balance if it's off and prevents you from accessing some features.
If you breed enough generations in the breeding simulation, the simulation stability will decrease, the dogs will become greatly mutated, and the music will become weird and unsettling.
Dogs can sometimes fight over things such as pieces of dirt, but no bad things can happen, and it doesnt happen very often. If you're insane enough you can command dogs to grab eachother (and even throw eachother, but the dogs will say "I don't want to do that" every time so it doesn't mean anything).
(Dog death can be disabled and their lifespans can be extended.) Upon dog death, their bodies break into pieces that can be eaten by other dogs, although given the game's cartoonish nature, it's not graphic at all.
the wobbledogs can become sad if you don't properly feed them or otherwise fail to take care of them, they will also howl when one of them dies if you have dog death on
There are no cats in the game. It is possible to breed a creature to look vaguely cat-like, and if you have death toggled on, then that creature will eventually pass on.
It depends on whatever is described as bugs:
Actual bugs: No, there aren't really actual bugs (even though there might be bugs in the spooky decorations like walls and floors).
Game bugs: The game has its fair share of glitches, so yes.
It is possible to breed dogs with extra limbs and heads. This is avoidable if you arent trying to get all of the achievments in the game, but doing so may limit the things you unlock. The art style is very cartoony and cutsey though, so this may not be a problem.
the game lets you decorates rooms so you could technically make stairs and put a dog at the top of them. The dogs are very bad at walking so it would probably fall down them.
there is a halloween-themed decoration item that is an eye in a jar of green goop. As well as an eyeball looking toy a dog might randomly dig up that the dogs will play with.
Ghosts were added in a fairly recent patch. To obtain them, you need to have a dog die and memorialize them. They will randomly spawn (for higher chances of spawning, make the room decorated like it's Halloween), but you can summon them if you'd like if one of them has already spawned. However, you can turn ghost spawning off, or dog death off for that matter.
Dogs v* a pink substance and a half-eaten chunk of food that must be dried up through player interaction. As of full release, you get an achievement for it.
Wobbledogs eat things with chewing noises, often bark (on command even), and can learn to howl. They sometimes vomit, but it uses an unrealistic sound effect.
The breeding in the game does not involve any mating - you splice the genes of two dogs together and insert them into an egg. You can do this with dogs that are "related".