Can the total number of each type of stamp be counted in a PDF?

Try going to the comments section, select your stamp, then press the "push pin" icon. Stamp away and you should be able to see the amount of comments for that particular stamp. You may also use the filter to select only stamps if there are other comments on the document. You will see the count in the comments viewing window.

Yes, this is possible using a script, either at the doc level or a page level, but the script needs to know the internal "name" of the stamp you want to count. This can be hard-coded into it, or it can be set up so that you select one such stamp first and then run the script and it looks for stamps of that same type.

I've developed similar tools for my clients in the past so if you're interested in hiring someone to do it for you, feel free to contact me privately via [try6767 at gmail.com] and we could discuss it further.