Those are some great questions on this topic. I like achievements, but it really depends on their quality, which varies a ton. Some achievements provide you with a very fun challenge you wouldn't otherwise do, and really add to the game's enjoyment and length. Others can be incredibly annoying and are doomed to be left incomplete by you forever, which is annoying in itself. If you don't like speedrunning for example, which I particularly hate, speedrun achievements are an instant and free "ugh, fuck this shit" the game gets. There are also retarded, unreasonable challenges like for instance taking a 40+ hours long game and putting an achievement like "complete the game without ever taking a hit" or some crap. The kind of achievement you can be sure will ruin and suck all the game's enjoyment out of you forever if you're stupid enough to embark on seriously attempting.
Multiplayer-only achievements in a game with a single player campaign are also shit. Even on multiplayer-only games I kinda dislike them to be honest. Team Fortress 2 for example has some notoriously ridiculous and random achievements that you just *know* people probably hop on a server with a friend or two to grab the achiev in a "scripted", artificial way.
"Infection" style achievements are the absolute bottom of the barrel though. Like Garry Mod's infamous "play on the same server as Garry", which they eventually turned into "play on the same server as someone who's ever played on the same server as Garry", so it's not that miserable to achieve but still an awful idea that should never be an official achievement in any game.
How important are achievements to you?
Not important, in the sense that I don't lose my mind if a game comes to Steam with no achievements feature, but as I said, I like them.
If a game has achievements, is that game only finished when 100% of the achievements are acquired?
Depends on the game. I'd say this applies for Sekiro for instance, which has really fair and obtainable achievements that essentially make you 100% the game's content, but it definitely doesn't apply for all games. Half Life 2's annoying and unreliable "find all 40+ lambda symbols spread across the game's walls" for instance can go suck a dick and I'd never consider it part of the game's true completion, just to give one example.
Are achievements a console mindset?
Don't think so.
Do you make sure a game doesn't have broken achievements before purchasing?
No but that's annoying to know, for sure.
Are there any achievements that you are proud of?
Hollow Knight's 112% completion, which I grabbed last month. Amazing game and great fun to get the achievement.
Do achievements have an effect on how you approach playing a game, such as looking up guides?
If I If already beat the game on my terms and I have already exhausted all possibilities there's no shame to check guides on how to grab the remaining achievements, especially if they're completely cryptic or doesn't even give any indication of what they're even about.