Can someone try to explain the use of some items/mechanics:

1) Mutagen

as far as I understand, the mutagen changes a random gene. So what do I use it for? If I'd want to change a gen, then it is either a pet with just a single flaw or a pet with many flaws. If a pet has just a single flaw, then changing a random gene is most likely bad for me as it is more likely to destroy a good gen than to actually target the bad one. And if a pet has many flaws, I won't even consider breeding it. So the mutagen looks pretty useless.

2)  neutering syrum

I have no idea why I would want to neuter one of my pets. Again, if it has many flaws, then I won't breed it but pond it instead. And if a pet has good genes, I'd like the chance to find a building that restores a breeding point- So what am I missing here?

3) invites

Most of the specialists that can be called with invites are pretty useless. So far, the only invite I ever used is the Geneticist. I don't think the devs would spend much time adding useless content so I must miss something here.

4) catching wild pets

I tried catching wild pets in the beginning but all pets I caught had so many bad genes that I never bred with any of them. Why would I want to catch wild pets if even the worst pets I can find in the pet auctions are better than them? Again, here is a major part of the game that I cannot understand why the devs would spend time working on it.

Thanks in advance for any comments/help.

Some people have huge piles of mutagen and are prepared to take risks, but I am not. I never use mutagen.

Some players breed excellent beasts to sell, but don't want the buyers breeding more. It's to protect their status and their sales. Not many players do it any more, as not many buyers really want beasts that can't breed.

The invites are tied into the wild-caught pets.

A lot of serious breeders, especially in the early days of this game, needed wild-caught pets to create new breeding lines. Wild pets are not related to any other pet in the game, meaning you generally can't get a degen baby from them. If you get a good beast off the pet trades, you might well find that they're related to the ideal male on the breeding offers that you've just found.

If you use the wild pet capture page a lot, most of the invites become very useful. If you have a massive breadth of genetic material available from the pet trades, however, and capturing wild pets isn't necessary for your breeding programme, then the invites are much less useful.

Travelling Magician, Financier, Strategist and Geneticist are in my view the most valuable invites.

Mutagens: I have actually used these! 

Basically, I was working on a breeding project starting from wild pets. Some genes are rare in wild pets, but mutagens have equal chances of giving any gene. So I took the wild pets with none of the genes I wanted and tried mutagens on them until I got something I needed for the project. (Obviously, this required a lot of wild pets and a lot of mutagens.) 

So: There's a use, but it's a weird precise situation. Most of the time, I stick with the majority opinion and don't use mutagens. 

Invites: In addition to the Genetecist, I find the Travelling Magician super useful during Tundra--being able to see whether there's a portal on a new island is extremely handy and can save a lot of headaches. I also use Trainer sometimes (see: hard to level up wild pets for breeding projects). 

Thanks a lot for your explanations.

I have another question, though:

How do I get more than three squads? I have seen several people with them but the military academy has max level of three as far as I know?

50 gold for a Mercenary Licence. Gives you one extra squad.