Each dog is different, but the best way to fix “accidents” is to prevent them from happening. Most dogs need much longer than a week and a half to understand what you want of them.
This is how I trained all three of my dogs:
The first couple of weeks they lived with me, I took them out about every hour during the day. A pain in the ass, yes, but it saved a lot of carpet scrubbing. Secondly, you HAVE to watch a new puppy constantly. Put him in a crate when you can’t supervise him. (Watch for sniffling and circling, sure signs a puppy’s about to let loose.)
At night, the puppy should sleep in his crate, preferably in your bedroom. If he cries, get up and let him out, but put him right back into the crate. At two months old, he’s a little young to be holding it all night.
Any time he whines during the day, take him out, even if he was just outside. That way, he’ll learn “whining=go out” and be able to signal to you when he needs to go.
After a few weeks of this, lengthen the time between outs, gradually stretching it to two hours, then a week or so later, three hours. What your goal is is to get the dog on a schedual. Mine go out at 9 AM, 11, 5 PM, 9 PM and 11 PM, and usually wait 'til that time before crying (unless they have an upset tummy.)
Do not punish the puppy for spots you find on the carpet. It’s okay to say “NO!” if you see the puppy squat, but if you wait even a few seconds, they don’t realize what you’re punishing them for. Dogs have a very weak grasp of cause and effect. Punishments must be IMMEDIATE in order to make sense. If you catch him squatting, say no and take him outside. When he squats outdoor, praise him.
As for when he can roam the house alone, that also depends on the dog. What I usually do is keep them in the same room with me, always where I can see them for the first few months. And yes, that means not even leaving them alone for fifteen seconds unless they’re in their crate. If you need to, put up baricades to keep him in your sight.
After the first couple of months start letting him roam into other rooms, but keep the area limited to where you’re always nearby. If no problems occur, you can gradually give him more freedom. But, remember, you still need to teach him what he’s allowed to chew on and what furniture he’s forbidden to use, etc.
Also remember that the puppy WANTS to please you, but doesn’t know exactly what his new “pack” requires of him. Raising a puppy is a lot like having a human baby at first. It takes a lot of time and effort, but they grow up a lot quicker.
Have patience with him. He’s still very much a baby. You wouldn’t expcet an infant to know after a month or two how to change their own diapers, would you? Of course not. He’s probably just left his mother, and he’s very confused.
There’s tons of advice out on the internet. Don’t assume any one method will work for your dog. You may have to make some adaptions before you find the right program for training. Also, always feel free to ask for advice here. There’s a devoted contingent of dog lovers who are always happy to give advice.
Good luck.