Since you requested that all of us help you out, I feel compelled to tell you what I know even though it may not be directly answering all your questions.

There is a reason why STRING limits the network size to 2000. Most biological networks are smaller than that - at least in a local sense, as ultimately everything is connected. Separately, visualizing a network of that size would be very difficult down the road. You did not tell us what were the parameters used to get this group of genes (log2FC, p-values), but I am guessing that you didn't set them very strictly. If you use more stringent values (e.g., set p-values to 0.01 instead or 0.05, or log2FC to 2 instead of 1), chances are that you will get fewer than 2000 proteins to work with. This will possibly cut down the false positives that always happen when working with a network of this size.

STRING is interfaced with many databases and is able to recognize most gene names, but that will never be perfect - and more so for 2000+ names. You can try to find synonyms for the genes that are not recognized, or submit protein sequences instead of names.