I'm sure it will feel very awkward but in a very short time this will be an entire year. and you have had to contact them,they show little to no interest in even a picture,let alone the dog its self. I can tell by how you write about this that you are a very caring person. so as I said I know it will feel awkward, but just tell them nicely but directly that you have decided you are keeping the dog. people that care that little shouldn't be pet owners. imagine how neglectful they would be with the dog in their care. they will most likely feel the need to put on a show of protest but that is all it will be,a show. and of course they won't like having the decision taken out of their hands. oh too bad, boo hoo. they abandoned the dog! please don't abandon that poor dog again by giving it to those people. 10.00$ does not buy a year of doggie care. you have earned that dog. stick to your guns,they won't.
This can be considered abandonment. This is because if the owners are still around, it should have taken a maximum of 6 months for them to start taking care of their dog. However, you need to talk to them and get to know their decision.
That depends on the original agreement. How long were you supposed to be watching the dog? Basically you should send them an email staying that they should have come and retrieved their dog by now. Say something like, if they don't get the dog within 30 days it will be considered abandoned and you will then be the rightful owner (or that you will rehome it, if that is your intention). Now if they don't respond in an email, you must send them a certified letter, return receipt requested. This is so that should they object in the future, you have a visible trail of how you handled/ agreed to everything.
In this case, I'd let sleeping dogs lie (could not resist, just couldn't!) They are not coming back for the dog, and I see no logic in raising the issue with them. I would consider her to be yours, but just in case they come back, keep precise records of all medical costs, food costs, and anything else you spend on it. Then if (which I would be shocked if they came back) they come back, insist that they pay your bill in full before you release the dog. I have no doubt this dog is yours--just let it be.
Have they been paying your regularly? Have they contacted you? Where are they? Unless they are out of the country, I would certainly think that they have abandoned the dog. Especially if they haven't paid you regularly. I would suggest contacting a lawyer or local Animal Control to be sure, just in case the people randomly show up one day and demand "their" dog back. Remember you can still go after them for unpaid sitting costs if you want to pursue that.