Progress scopes with an A serial number prefix were made in 1942 and early 1943. Ones with a Cyrillic B prefix like yours were made from early 1943 into 1944. From the low serial number of yours, it was made pretty early in 1943.
Post-war Soviet era bases and mounts are generally unmarked.
Yes, the wood needs to be taken down some so that it does not contact the base anywhere. You don't need a huge gap, but you do need a little space. If this was one of the Century rifles, there would have been the proper spacing around the base so it didn't contact the stock.
Actually, there is nothing wrong with the Century snipers - some of them were well done rifles and good shooters. They have their own unique place in Mosin history. Many of their earlier ones had original WW2 scopes, mounts, and bases. Later Century got in a lot of original WW2 scopes with post-war mount setups and used these to put together replica snipers. Of course, they also sold these scope and mount setups separately, and a lot of people used them to build their own sniper replicas.
I used to buy these Century replica snipers real cheap, steal the original scope and mount setups off them, and then re-sell the rifles. This gave me nice scopes to use on my sniper restoration projects, and someone else who had a scope setup laying around would put it on the rifle and return it to replica sniper configuration. No way of knowing for sure, but yours might be something like this too.
Are the locating pin holes drilled into the receiver, and if so, are the pins in the holes in the base?
I wouldn't try to return this one to original configuration - just enjoy it as a nice replica sniper.
Of course, if you ever came across a nice original sniper that needed a scope setup, like the ones Samco imported from Yugoslavia or the Soviet-Polish-Albanian "asterisked" rifles Century imported back around 2000, you could put this scope setup on it, and then get one of the current replica PU scope setups to use on this rifle.