Using other pic microcontrollers instead of the PIC12C508 (arcade monitor protection circuit
Do you have the source code for this project? The PIC12C508 is very basic compared to the other two microcontrollers you mentioned, and this program should be easy to duplicate on the newer chip if we could see the code.

It's very unlikely that the hex file you provided will work on the chips you have access to. In theory you could disassemble that hex into assembly but I have no experience in that process.

Edit: Found this on the first page linked. This is a fairly simple program. It's not well documented but it should work with little modification.

Your question about pins can be answered by finding the datasheets for the PICs in question. The easiest way to do that is to Google the full name of the PIC and choose a link that starts with "www.microchip.com". Here is the datasheet for the PIC12F510. All Microchip PIC datasheets follow a similar format. There is usually a cover page, then a page with legal information, then a device overview page(PDF page 3). The next page of this datasheet has the pinout diagrams for the different packages. You are probably using the 8 pin PDIP package. 

Comparing the datasheets tells us that the PIC12C509 and the PIC12F510 are pin-compatible.

So yes, you can use the 12F510 in this project. Moving forward, what do you know about programming a PIC? Do you already have the tools to program a PIC?