With the desktop download, Once I paid and I have an issue downloading it to my computer, do I still have access to the software to download it somewhere else?

Yes you can download it again to another Mac that is newer.  Sorry your Mac is too old.  Mac System Requirements     https://ttlc.intuit.com/questions/1899173-minimum-system-requirements-for-turbotax-mac-software

But if you want to use a Windows computer you need to call in and get the Windows download.
Or you can use the Online web version.  But then you need to pay for online separately and  get a refund for the Desktop program.