there's a lot of questions in here. I'll quickly hit on them.
Is Square's card reader hardware compatible with any register system?
No, Square hardware is only compatible with Square's software.
Is there a benefit of getting the whole register system with card readers?
Yes, you get the Square Point OF Sale software to go with the readers to get all your sales data that is saved to the Square Cloud servers that you can access from anywhere and download to use in excel.
What is cost per register and is the register compatible with other inventory and accounting software?
This can be a little or a lot depending on what you are trying to do. you can use an existing phone with a $49 chip reader and be ready to take payments, or you can set up an iPad and cash drawer and printer to be a full cash register setup for around $500. Inventory is built in to Square and is available to export but I wouldn't say it is compatible without some messing around. people do use Quickbooks for accounting and there are other software suites also. see this link https://squareup.com/us/en/app-marketplace?category=accounting-and-tax
