The primary issue with excess fat (and remember it's not purely the butterfat % that matters, but the ratio of fat:protein) is that the fat gradually breaks down through lipolysis. This process is good to a point as it results in new flavors - cheese gains flavor as it ages - but at a certain point it becomes excessive and results in off flavors.
You may experience bitter, rancid, and pineapple-y/bitey/butyric acid flavors (ok, not sure if these are from lipolysis or proteolysis but you get the idea).
If a cheese is properly made and salted adequately it generally will withstand a decent amount of aging if the butterfat is reasonable (up to about 0.8:1 protein: butterfat).
Lower salt cheeses (like some Alpines) or very long aged wheels (think Parm) need less butterfat as those off flavors will develop more quickly.
So if you have high fat/low protein goat milk, the milk will be great for soft fresh cheeses, soft ripened cheeses and semi-soft/shorter aged wheels like Colby, Tomme, etc.
Cheddar would really depend on what your p:bf ratio is - you can get a components test on the milk or just give it a go and find out.
Parm simply doesn't achieve the proper texture with whole milk but you can still make a nice grana style cheese.
The Slavic Beauty separators aren't super expensive and they work really well. Or add some store-bought cow skim milk
