You can use this website to calculate the string size: http://www.strothers.com/string_choice.html . It tends to be a little on the light side. When I did the calculations, I came up with D = 0.018, A = 0.012, and d = 0.0090. I would suggest trying 0.020, 0.014, and 0.010. When you take the strings off, you can use a small block of wood and hammer to knock the pegs out if they do not come out on their own. You can clean them off with a mild soap. Murphy's Oil Soap works well. A small amount of peg dope available from a violin or music shop can help lubricate the pegs for easy turning and grip. I also use the Murphy's for cleaning the entire instrument. Where the scratches are, you might need to sand them out and refinish the area. It depends upon how much you want go get rid of them. Best wishes on your project and the new (to you) dulcimer.
Ken
"The dulcimer sings a sweet song."
It sounds like you have a solid plan for maintaining and improving your dulcimer! Using peg dope for lubrication and Murphy's Oil Soap for cleaning are great practices to keep your instrument in good condition. Sanding out scratches and refinishing areas as needed can also help restore its appearance. Best wishes with your project, and I hope your dulcimer continues to bring you joy and beautiful music!