Are you looking to have 3 females or 3 males? Three can work well with females, but it's rare for 3 boars to get along successfully. Boars are generally best kept in pairs.
It's possible to bond a pair of sows with an existing sow, but I personally have no experience in doing so. Hopefully someone with more experience will come along and have some advice for you on how to do it. I do know that whether or not the introduction is successful depends in a large part on the individual personalities of the pigs. It's not unusual for the dynamic between a bonded pair to change once a third pig is introduced.
As for your questions:
1.) Again, I think it really depends on the personality of the pig as to whether she will be happier as part of a small group or just with a single friend. In general, I think most pigs are probably happier as part of a group? But again, I have no experience here; I've only ever kept 2 at a time. But most pigs do just fine when kept in pairs.
2.) More animals will increase the mess (obviously), but as long as the cage is large enough for 3 animals, it shouldn't really start to smell much faster or need to be cleaned more often than if only 2 lived in it. Although if you use loose bedding, you might have to throw more bedding out with each spot clean because of increased mess.
3.) You have to account for increase in pellet food, veggies, hay and possibly a small increase in bedding of course, but the most significant cost increase of adding a third animal is going to be in the veterinary fund. Exotic vets are definitely not cheap, and I would highly recommend that you have a stable veterinary fund ready BEFORE you increase your herd. The amount of money you choose to put aside for this may depend on things like where you live, what exotic vets are available in your area and what their prices run. In general though, most people would recommend putting aside AT LEAST several hundred dollars PER PIG to cover the cost of a veterinary emergency. My own vet fund for my pigs had about $500 per pig MINIMUM at any one time, but ideally, I liked to have closer to $1000 per pig.
To answer your overall question of whether or not 3 pigs cost more than just 2 pigs plus one more: no, there should not be as much of a cost increase with adding additional animals once you already have one. This is because even though you have to account for more food, more hay, more veggies, increased vet fund, etc, the pigs will be sharing a lot of things that made the startup cost of getting your first pig so expensive. For example, they will be sharing a cage and bedding, so you won't actually spend a lot more on bedding with 3 pigs together in the same space.
