I wouldn't mess with the sight hood. Sliding it on and off will wear the blacking on the barrel. I would instead work on removing the original sight blade and storing it with the others and having a replacement made that can be replaced if it gets broken.

I'd definitely send it to a gunsmith knowledgeable in restoration such as Lon Paul for an assessment and stabilization. Complete disassembly and evaluation with cleaning and conservation of the the wood and metal is time and money well spent.

If you lay the groundwork and take care of the details you can put hundreds of rounds through it and I even take it on safari if that's in the cards for you, but if it was my rifle I would definitely consult professionals and proceed in a stepwise manner.

I would not just clean the bore and shoot it.