> Will LTE continue to work when 6G rolls out in 3 years?

A significant change that came with LTE is the use of OFDM as the base of the air interface, which means that cellular providers no longer need to allocate large segments of spectrum to one protocol or another. Towers that were upgraded from 4G to 5G didn't need to disable 4G or consistently split their spectrum between the two they way they did when upgrading from 2G to 3G or 3G to 4G, they can dynamically allocate subchannels in each mode based on the immediate needs of their users.

It seems unlikely that whatever technologies come in the future will regress in this way, so there shouldn't be a need for a hard cutoff at any point and I'd expect LTE devices to keep working for many generations to come.

Probably for a while - lots of IoT uses it, especially with the push away from 3G.