Ok be patient with me here. lol
I used to have a shopsite store and was told it could realtime inventory updating and when it came down to it it couldn't do what I wanted it to do so I wasted a lot of money on getting a shopsite website built.
Can shopsite templates be easily and cheaply converted to magento?
I don't believe it can be easily converted to Magento.
You may need to consult with a developer to see if it is doable but I don't think the cost will be cheap.
Unfortunately, there is no easy direct way to take a ShopSite template and convert it to a Magento one. The two formats are not compatible.
If you have a unique custom look / design, you would likely need to hire a Magento design/development firm to re-create that look in Magento. Otherwise, you could look at the various pre-built Magento templates out there to see if one comes close to the look you'd like.
What if I have the original psd files and html5 files? Would it be easier then?
Yes, that would be helpful for a Magento designer to help create a custom theme for your site.