THE GOOD COMPANIONS - Nikki White
Cavies are herd animals and should not be housed alone if possible. Exceptions would be stud boars between matings if he won't accept his young sons or they won't accept him, and long-haired show cavies because of the problem of barbering. Sows usually get along fine together. Even boars can be housed together. I know of one person who has a whole lot of them in a big enclosure up the back of his yard. The best combination is usually a father and son(s) or at least an older boar and a younger one but I have housed adult unrelated boars (not even the same breed) together successfully. The trick is to make sure there isn't even a whiff of a sow around.
When introduced, there will often be the usual purring, teeth-chattering and bum-waving until they have sorted out who is top pig. This applies to sows as well as boars. If things turn nasty and they fight, then they will have to be separated. Fortunately, this doesn't happen often as cavies tend to avoid fights, one will usually back down before a blow is struck. If they do fight, don't put your hand in to stop them. You could come away with a nasty bite. Throw a towel over them or squirt them with a pump-action water bottle.
Sows kept together are good at looking after each other's babies, even those who have never been mothers will help. Sometimes a pregnant sow will 'hijack' the baby of a sow who has recently given birth but will return the baby when she gives birth herself.
Other animals
Mice
No definitely not. They can carry a disease like mouse fever. Their droppings can foul the food and water, again causing disease.
Rats
Again, no. They have different dietary requirements (being omnivores while cavies are strictly herbivores) and they can carry disease.
Cats and Dogs
Obviously they don't belong in the cage with the cavies and their idea of play might injure the cavies considering the size difference. However, they can be introduced properly with care and allowed to play together under supervision. I do not recommend having free-range cavies in the house, especially with larger pets like these.
As I have never owned a dog, I can only talk about cats. I covered the metal top of the indoor cage with flyscreen wire to keep out inquisitive paws as the cats, an Abyssinian and a Somali, were kittens at the time. However, they had a habit of lying on top of the cage and staring at the cavies. As they grew older, especially when I moved in some less active cavies, they seemed to lose interest. The current crop of indoor cavies chase each other around and don't even have a flyscreen wire covering on top yet don't attract the cats' attention as they are now two years old though the young Bengal has shown some interest in lying beside one of the cages occasionally.
My late Siamese took a keen interest in a Dark-Eyed White sow I had when she was little and that rat-like stage. But once she filled out into the usual cobby cavy shape, he just ignored her. My friend's Siamese crossbreed made the mistake of poking his paw into her Self Chocolate sow's cage and got it bitten. Thick as a brick, he tried it again later and got the same result.
Basically, over time, the cats accept the cavies as part of the household and of no more interest than a houseplant or the house bunnies. However, I believe they should be supervised if the cavy is brought out of his cage to play with the cat(s) to prevent any rough stuff. You do see photos of cats with long-suffering expressions and a cavy parked on their heads. A cat-breeding friend told me of a kitten who was abandoned by its mother and who was raised by a guinea pig. That would be the cat who calls for dinner by going "Wheek, wheek, wheek!", then.
Rabbits
This is a vexed question. While many pet books recommend them as companions and many pet shops house them together, an increasing number of people and books, especially in the UK, are firmly against it.
The reasons against it include: Rabbits, being bigger, can inadvertently harm a cavy when bounding around, they may try to mate with a cavy, or kick out at it if annoyed and they have powerful back legs. (A lot of these 'no-rabbit' bods make out the rabbit is always the villain but I know of cavies who bite rabbits, bail them up and won't let them into the hutch or try to mate with them. So it's a two-way street.). Rabbits have somewhat different dietary requirements - if they eat as much green stuff as cavies they could get digestive upsets; rabbits can in some circumstances give cavies bordetella pneumonia, a disease that is usually non-pathogenic in rabbits but fatal in cavies. However, this is likely only to occur in overcrowded, dirty conditions. Cavies cannot get any of the fatal rabbit diseases like myxomatosis or rabbit calicivirus (RVHD).
On the other hand: Rabbits make a nice, soft warm companion to sleep on, can keep cavies from the cold and being a bit bigger can (and do) defend them. To avoid some of the problems listed above, choose only a Dwarf breed similar in size or not too much bigger than a guinea pig (Netherland Dwarf, Dwarf, Mini or Cashmere Lop or a Dutch). Medium rabbits like the Satin or Rex are not suitable and forget the giant breeds like the British Giant or Flemish Giant. They'd squash a cat or a small dog. A placid natured rabbit is best such as a Dutch or any of the Lops.
Commonsense in feeding should avoid digestive problems. Giving a couple of slices of carrot or cucumber a day to the rabbit is not going to upset it, just make sure it is more or less the same amount as the cavy gets to avoid jealousy and "thieving bunny" syndrome. Grass if available is always good for both and doesn't cause any problems.
I have run the two species together in the past and generally have had no problems. It isn't something I make a practice of and until recently hadn't done it for about three or four years and then only with pets. I have had Lops and a Dutch crossbreed in with a pair of sows and other similar combinations. Whatever the case, I found the cavies tended to take over the hutch leaving the rabbits to roost on the roof. The cavies liked to curl up and sleep on the does who were lying in the winter sun. And if they felt threatened, they would crawl under the nearest Lop, knowing s/he could deal, which was a bit sad when the Lop was only a kitten and not much bigger than the cavy.
Most recently, I had to separate some boars, one of whom was bullying the others (Kintaro, a mouthy Golden Agouti Rex baby). So I put him in with the Dutch rabbit in the cage underneath. This seems to have worked well as Maximus, the rabbit, is too big for Kintaro to bully, in fact he just ignores his new companion if he get stroppy and you haven't been ignored until you have been ignored by a rabbit. Kintaro seems to have calmed down and brought Maximus out of his shell. Maximus obviously has ambitions of showing Kintaro as he keeps grooming him.
I always tell people who ask about this combination that it is possible, given the caveats, but they should be prepared to house them separately if things don't work out. They should watch them carefully to make sure they don't fight. Some rabbits won't have a bar of guinea pigs and there are cavies who can't stand rabbits and bite them.