Throw Away Pokemon (without checking IVs)

No.
I value every pokemon and i like to keep perfect pokemons.
Its catch em all kind of feeling

I don't check IV on anything that requires 12 or 25 candy to evolve, not including Eevee and Mons with a 2nd evolution. So I don't check pidgey, caterpie, Merrill, rattata, etc. This is because in my mind they exist mainly to evolve and transfer.

I do check Mons like Rapidash, but only because they are uncommon enough and take 50 candies, so that I need SOME way to determine which one to evolve.

But I also go off of CP. For a really crappy Mon like krabby, I may just evolve the highest CP one. What is the point of having a 89% Kingler with 1500cp when I can have a 50% with 2000cp? The 2000cp will be much better to attack a gym, then I will transfer him.

Pidgey, Weedle, Rattata, Caterpie, Sentret, Ledyba, Zubat...etc

In general as long as I have a final evolution of a useless pokemon that has the optimal moves, good IV and decent level, I just batch transfer the rest.

For obvious reasons checking magikarp is a good idea even if you have a decent gyarados

I'd keep perfect Pokemon even if they're useless just for personal collection purposes.

I already have a few high IV pidgey, but I generally check them. Then I transfer the ones with useless moves. Same for the new trash Pokemon (introduced in Gen2) until I get a few decent ones.
I pass sometimes, for very low level Pokemon.
Sometimes I also feel that I'm getting a bit tired of checking IV

I only check iv's inside the game app for those mon's I am interested in. I also use 3rd party iv checker to check iv's of my new catches, at that time I will keep 100% iv trash mon's if I happen not to have transferred them earlier.

If it is something that I might use, I start with in-game appraisal to weed out the worst. Something that might be an attacker and defender I want to be at least in the 82%+ IV category. Something that will evolve to a CP range good for prestige I want to have high stamina IV, regardless of total IV percentage category.

For rarer pokemon, I will use Poke Genie to refine my selections. For more common pokemon that I have a good amount of candy stockpiled, I often only further check IVs after evolving if they have good movesets.

I don't usually check pokemon that have no possible use after evolving. After mass evolving though, I do go through and then check those that have movesets that I know are good. Kinda backwards, but I'm only going to keep a useless pokemon as a trophy if it has good IVs and moveset... doesn't really matter which I check first.

If the mon is below lvl 20 I don't want to know if it has great IVs or not, so I'll blindly transfer it.
There are only a handful of mons that I will actually check the appraisal before transferring no matter how low the cp.

Have to admit my OCD makes me at least appraise everything, just because I'm looking for trophies - my pattern is catch, appraise, fast-tap through rest of appraisal for anything that isn't top-tier (and usually transfer right away). If it is 15 in 2 stats then I will usually append eg. -HD if it's 15 for HP and defense), and snap a screenshot for PokeGenie to use later, I tend to keep 15/15/14 mons too, possible use for breeding. Not sure if I will keep doing this tbh since outside of defenders IV just doesn't have much impact on my game play.