Housebreaking I can help you with. When I trained dogs professionally, one of my services was to take puppies for a month (or longer) to get them reliably housebroken. It cost a lot of money. Not because housebreaking is hard; because fixing owner mistakes is hard.
I will assume that Cora is crate trained.
I am not clear what it is that Cora is doing “wrong”? How old is she now? When housebreaking a young puppy, peeing or pooping anywhere outside is wonderful. Later, you can guide her to more specific areas for potty, if that’s what you want. But, to get a puppy to understand a) outside, and b) only certain places outside is going to confuse her.
The potty in the garage thing must stop. You are teaching Cora to potty indoors. A puppy has no way to understand that a garage indoors is different than a house indoors. Inside is inside to Cora. More confusion.
What do you mean you’re taking Cora out at the ‘right times’?
If a puppy is not 100% reliable in the house, she needs to be crated whenever you can’t watch her every single second. That is not hyperbole. Your job is to prevent her from ever making a mistake. Every time a puppy potties in the house, she learns she can potty in the house.
You need the reflexes of a military test pilot for this. It is really hard work. But, it passes.
A baby (under four months) needs to be taken out every two hours, with the exception of overnight. Also after play; sometimes during play, after eating, after a nap, when she comes out of her crate, and a few hundred other random times each day. Getting her outside often enough really makes this easier. Don’t drag her, scoop her up and carry her. Never wait for her to come along. It may be too late. Be decisive.
It’s all about prevention. Don’t set her up to fail. Don’t confuse her. Pick a spot she seems to gravitate toward and make that Cora’s designated potty for now. Bring her straight to that site each and every time you take her out. Her own pee scent will really help—it’s a trigger. It’s ok to pick up the poop. There will be enough “remnants” to remind her.
Give this at least 30 days.
What are the issues with other dogs and people?
Grooming should not be at all traumatic, if the groomer is a good one. Word of mouth is the only way to find a good groomer. You can also ask the staff at your vet’s office.