How can I test to see if I can download windows 10 on this computer

Assuming your Windows 7 or 8 is legally Activated, the linked Upgrade Assistant App also runs on Windows 7. And W 10 will test your PC's compatibility during the Installation, too. 

Windows Vista, or XP and older, PC's do not qualify for the free W 10 Upgrade, nor do PIRATED Versions of Windows. check if the strength of your computer matches with the specifications of your computer.

if I recall right, win 10 upgrades is free for win 7 and 8 uses until July of this year. If your computer is compatible, an 'upgrade to windows 10' icon should appear in your system tray. there is no way to download win 10 without that icon. the upgrade process is nearly fully automatic, it will determine compatibility on its own. 

if you cannot get that icon to appear, then your system is not eligible for a free upgrade (a $200 value) and you'll have to buy it from a retailer. win 10 system requirements here. https://www.microsoft.com/en-us/windows/...

buying it from a retailer you'll have the option to download vs physical copy, the download is about 4 GB in size, downloading can have problems if your internet isn't fast, stable or reliable, a physical copy has an addition advantage that if you need the OS disk for any reason you don't need to re-download, burn iso to DVD to use.

Source(s):ps, fast internet isn't required for download, but if it's too slow there's a good chance of repeated time out connections and permanently stuck partial downloads.