1: Shaking a new bottle of nic solution before first use is recommended, and of course shaking a finished liquid after mixing is a must. Aside from that, no, I don't usually see people shake their liquids before each use. You must not be watching the same videos I am. You shake it initially to make sure it's all evenly mixed, but once that's done, it shouldn't "unmix" itself.
2: Shaking a sealed bottle, won't introduce any more oxygen than what was already in the bottle.
3: Here's a quote from our resident chemist
@Kurt about oxidation in sealed glass:
If you feel better by purging the headspace air with N2 or Ar to rid it of O2, ok. However, I think it is a negligible effect. 3 mL of headspace air at 25% O2 will give enough O2 to at most react with about 5 mg of nicotine. And that is only if it all reacts. VG naturally contains about 2 mg of O2 per liter, which if it all reacts will oxidize about 0.38 mg of nic...per liter of VG.
Note that that's not 5mg of nicotine
per ml, but 5mg
overall. So in a 120ml bottle with 100mg/ml nicotine, that's 12,000mg of nicotine. If 5mg of that reacts with oxygen, then there's still 11,995mg remaining, at a strength of 11,995/120 or 99.958333... mg/ml. In a 1L bottle, it would be 99.995mg/ml instead of 100 even. Once you mix that down to a finished liquid, I doubt that even the most sophisticated of tasters would be able to tell the difference.