I'm surprised a rescue sent him home without all that stuff already done. That's odd.
Don't freak out over cost -- we can probably help you figure out how to get it all done for a lot less. However, what's your plan for future vet care costs? This can be a very expensive breed to own. Most of us who own this breed have known too many $1,000+ vet bills over our lives. Good pet insurance for illness and injury costs about $50/mo (through Healthy Paws, for example) -- but you still have to be able to front the cost on a credit card while you wait a few weeks for reimbursement.
For neutering...ask the rescue for a list of low-cost speuter resources -- or simply Google "low-cost neuter clinic [your town/county]." These speuter clinics are usually concentrated in larger towns, but it's common to be able to get it done for $100 or less at partially grant-funded places (and some of them may even do it for free, depending on your income). Those places also usually do very low-cost annual vaxing (you can also find low-cost vaxing at feed store clinics and at THRIVE clinics, located in certain Petco stores).
In CA, you can also use very inexpensive HW prevention because it works well against the kind of Heartworms there (and No Cal does have heartworms, unfortunately) -- you can spend as little as $5/mo ($30/6-pack) if you buy a generic with the same active ingredient as Heartguard online with your vet's RX (ask for Iverheart Plus or Triheart plus).
Check out this thread for useful info on bringing down cost of basic vet care:
As for cats, our rescue's cat-owning fosters make the cats off-limits and don't allow dogs to chase cats *at all* (even in play). The cats decide when they want to make contact with a calm dog. Interestingly, quite a few of the GSDs we've taken in seem to really love cats -- letting cats groom them and even "make biscuits" on their fur, etc. A few need to be whacked by a cat one time to stop pestering the cat and learn respect. A minority will chase them and never can be safe unsupervised with them. Most can be taught to at least ignore them. Don't assume the worst yet, but be very vigilant and firm.
A two-week shut down would be very beneficial to help the dog adjust to your home, your critters, and your daily rhythm -- you can find lots of threads about it in the archives.