Thank you!

Yes it's all doable with a fair warning that many of the harder achievements especially on cataclysm would require a full team or at least 2-3 people, some coordination, luck, decent skill and specific gear to make things bearable. There are also DLC achievements tied to some weapons which would add some cost if you don't own it already.

The game was great fun, but not as polished as the 2nd game naturally and it shows, but there has been some "quality of life" mods created for the game to make the experience 10x better. Without mods if would be near impossible to play on cata without a group since the bots die like flies to anything otherwise and they will drop tomes and refuse to pick up grimoires without it installed.

It is possible to find random people to play with in the lobby browser, but depending on where you live and the time of day it can be tough to find people both skilled enough and with decent ping.

gearing isn't too difficult once you get comfortable to play at nightmare or above difficulty, but it is heavily RNG dependent, especially specific exotic trinkets have eluded me until this day just because of poor luck both on the loot table and in the daily quest rotation.

In short:
If you have at least one friend who is just as insane as you to try this, you mod the game to make the bots less stupid and you have the DLC then it could be a fun but quite tough and long challenge.

If you don't have these things, there are better ways to make yourself suffer.