1- creature positions and movement (e.g. TotM vs battlemat, step-by-step movement around obstacles, counting diagonals x1.5)
Both. My players, however, like tactical combat, so we use battlemaps more often than not. Currently I am running a massive dungeon crawl, so there a map up on the VTT with tokens for players and monsters pretty much the entire time. Some random encounters or more simple encounters I'll do modified theater of the mind, but they are still looking at a map, I just don't bother putting tokens down.
When I did Curse of Strahd, I used TOTM a lot more. But I still bought the Schley maps and printed them on a large format printer for use a battlemaps with minis for major encounters.
My first campaign in 5e was homebrew and I did a lot more TOTM and when a battlemap was needed I usually used the Chessex battle map that crudely drew terrain / room features on.
2- distances and weapon/spell ranges (e.g. measuring exact ranges vs ballparking near/far)
I play it RAW. On the VTT it is easy with the area-of-effect templates and distance rulers. When I was using physical battlemaps and minis, I had some plastic templates, but found them more of a chore than a help and would just use measure sticks, when it was hard to eyeball it.
3- elevation, angles and directions (e.g. adjudicating cover carefully from different directions)
Currently, yes, but it is easy in a dungeon. When I did a lot more wilderness and outside ruins and city adventures, I would often guesstimate.
4- terrain and obstacles (e.g. ignoring vs taking small objects on the ground into account, like a chair)
I use difficult terrain rules and may call for athletics or acrobatic checks. But I'm forgiving. Players are heroic and I don't want to slow the game down for every little possible opportunity for spraining an ankle.
5- lesser actions on someone's turn (e.g. handwaving/enforcing object interaction rules, switching an object between hands)
I don't bother with switching hands, am pretty loose about picking something up, but rather strict about getting things out of your pack, bag of holding, etc. during combat.
6- spells somatic/material components (e.g. checking against free/occupied hands)
Pretty much all magic users in my current campaign use a focus. For the most part, I don't pay too much attention unless players start getting ridiculous with it.
7- light sources and visibility (e.g. keeping track of exact areas of bright light, dim light, darkness)
I guess not really because once darkvision comes into play, I just roughly keep in mind how far characters can see. Also, I'm not running a VTT with lighting effects where each player only sees the what their character could see based on vision and lighting rules. Everyone pretty much sees what the deep gnome can see.
8- any particular tactical area (e.g. hiding in combat, flanking, facing)
For my current campaign, I don't use optional tactical rules from the DMG. I did in prior campaigns, but characters in 5e already are so powerful that it felt like it was making encounters too easy. I prefer to make feats and special abilities like pack tactics mean more by not allowing everyone flanking.
Hiding in combat, I do follow RAW. Not doing do would be taking away a lot of the fun for players playing rogues.