There are a couple of types of mold that are hazardous to your health. There are thousands/millions of types that are not. Someone at Biology.se* might be able to identify by picture whether or not your mold is hazardous, but the best bet would be to have it tested.
To kill the mold, a dilute solution of bleach should do the job. Spray down all visible areas of mold, both inside and outside (if accessible).
Cost effectiveness of having someone else do it for you is entirely up to you to determine. We don't know where you live so we can't give you a quote or even tell you if such a service is available where you live.
As you've noted, you see mold even in new installations, so replacing the AC unit is not a surefire way of not getting mold in it again. The only real reason to replace the unit is if that ends up being less expensive than repairing a broken unit again.
To prevent future mold growth, clean the unit regularly. That dilute bleach solution should do the job. Of course, if it's not a hazardous mold, then it's a purely cosmetic decision to clean it or not and that is, again, up to you.
NOTE: It strikes me as very odd that the landlord would have allowed a previous tenant to install a through-the-wall AC unit since that is a permanent modification to the building requiring cutting a hole in the wall, modifying framing, water-proofing the intrusion, new wiring, etc. Do any of the other apartments have a similar installation? You may want to double check with the landlord to see if he was aware that the change was made (before it happened), when it was made (i.e. just how old is this unit), and that it met with his approval. You may want to review your your lease agreement to see what's required to make this kind of permanent change to your apartment.
Also, if you replace the AC unit on your dime, you're investing into the landlord's property. That's very considerate of you, since you'd most likely move out in a year or two leaving behind a brand new AC unit worth thousands of (local currency units). If it were me, I would give this some very serious though...
*Reading through their On/Off-topic listings, I'm not sure if this would be on topic there or not, you may want to join the stack then ask in chat if it would be appropriate.
