do you think Web browsers will be supported on High Sierra for at least a few more years?
Pure guesswork, but a qualified yes. For example, I had to upgrade my mother's 2009 iMac from 10.11(El Capitan from 2015) to 10.13 (High Sierra from 2017) because she was reporting that Safari spat the dummy on a lot of websites important to her. I had a look and found that under 10.11 the browser was no longer capable of doing what she needed, chiefly internet banking. After the upgrade to 10.13 - which like your sister's Mac is as far as it could go - it was motoring along. However, that was one year ago...
Safari depends on technologies in macOS, but third-party browsers are less reliant. So if the OS can't be updated to adapt to changes then Safari falls behind too. As you saw with Chrome, all browsers have their sunset clause on any system too and, dare I say it, start from behind Safari (IMO) but gain when Safari can't be updated yet can hang around for a bit longer on an old OS.
Considering all that, the life of a computer was considered to be about five years (three in businesses), but Macs generally last much longer, certainly longer than a cheap PC box. The longevity bequeaths the problem of out of date hardware, though for many people who don't thrash their machine it isn't a great issue. Apple historically, and still now, pushes the OS software all the time and is not afraid to escort old hardware out of the building if necessary when it comes to a new OS.
Time Machine is one of the unsung heroes in macOS. Not only is it a painless backup regime built into the OS, but the migration of data to a new Mac is equally painless. It's set and forget, open a can or two of Kilkenny and come back in an hour and job done. It's never let me down with any of the multitude of my family's Macs and I've been using it since it was introduced into the system in 2007.
When your sister buys an external drive (SSD preferably) it will probably be formatted for Windows. It would be best to reformat it for Macintosh. Apple's guide here will see you right. She will have to use Mac OS Extended as the format. This is the old file system used by macOS 10.12 or earlier but will still be fine to use after the upgrade. I'm using that format on a drive I've had for a while for my Time Machine backups. I'll reformat it at some point, but it's okay for now.
These forums are for all users, not just the experienced users. Return for any question you might have and you'll get a response. Some of them might even be useful : )
Slán
David