Try a different browser
No, it's still possible to buy Lightroom standalone as an upgrade. You need to contact Adobe directly (what you did...), however, if there are problems.
Here is, how it should work: Download and install Lightroom 6 (Single App license)
The upgrade is indeed cheeper, but you get less: no Photoshop and you need to wait and upgrade again for new features.
The browser looks like Firefox and should work, if the program is updated to the latest version. But yes, that may be a good idea...
Instead of my usual browser, Firefox, I used Internet Explorer and everything went well.
Yes, "Checkout" does indeed not work using (a recent version of) Firefox.