1. A Shadowland (SL) account [an account where one bought the Shadowland expansion] will continue to be a SL account [having some access to SL zones at least], whether one's sub is off or not.
Level 20s on SL accounts may have trouble getting to a SL zone though due to level requirements [quests that lead one to the next set of zones often have a level requirement], though if it's like previous expansions some time ago, that SL-vet-20 could be summoned to one of those most-current zones through a mage portal or warlock summon. Non-SL accounts of course can't access those zones/etc., though characters from non-SL accounts can still buy SL-enchants/consumables/etc. from other players (if one can use those, which is another question [based on level and ilvl requirements]).
2. No, level 20s don't get locked when one lets their subscription lapse. Only characters level 21 and greater get locked when a sub lapses.
3. Once you let your sub lapse, the starter restrictions kick in on your ex-sub account, which includes no ability to use mail or AH [that includes no ability to see AH listings either].
If you ever want to get an idea of the AH market for your server while your sub is down, you can reference theunderminejournal, among other possible sources outside WoW.
In the past (not sure if this is still the case), the only time a "starter/vet" can access mail is if it's from a blizz GM or from item restoration [I'll need to double-check that item restoration part. Was a possibility some time ago, though I'm unsure if it still is).
Minor detail to note: if a F2P/Vet 20 has more than the starter gold cap [which is unclear as of this time, as F2Ps currently can hold more than 10g. To me, it's unclear if it's a bug or intended], or has a profession level greater than 100 on a profession, neither the gold or profession level goes down once the restrictions kick in. Gold goes down over time though due to repair bills/etc. if it's above the starter gold cap.
Long time ago though, I had a 255 profession go down to 100 when I did the DMF profession dailies [or similar profession dailies]. I don't know if that's still a situation or not though. Re-subbing didn't restore that profession level either. That said, there's many Vet characters here who, with sub off, continue to keep the high engineering profession levels to use gears/trinkets/bombs/etc. with no hiccups like that.
4. Currently, F2Ps can access Vanilla to BFA zones, with no mage or warlock assistance needed now.
Historically, if current expansion is expansion #N, F2Ps were able to access expansion #(N-1) zones [though mage or warlock summon was needed sometimes, like such assistance (before SL now) was needed for F2Ps to reach Outland and Pandaria, while F2Ps were able to travel to Northrend, WoD, and Legion zones by themselves]. Sometimes access to #(N-1) zones was delayed until expansion #(N-1) was added to the battlechest.
