I have indeed made vinegar for decades. I've found red wine vinegar to be easy, and white wine or dessert wine to be inexplicably challenging. I've nevertheless found interesting uses for dessert wine vinegar, such as making batches of tamarind paste for Indian cooking. One can't easily place what I did, but it makes a difference.
All facts about vinegar-making are in dispute, so read critically and experiment. One wants a good mother, or it's enough to drop in an ember from the KK. One needs to neutralize sulfites in wine by adding 1/2 tsp hydrogen peroxide per bottle (that H2O2 bottle in the cupboard is probably flat), or it doesn't matter.
I used to use a beer-making glass carboy, with the opening covered with cheeesecloth. I have since fallen in love with French cooperage barrels targeting vinegar; I have four 3L barrels, two on each coast. One can draw vinegar as needed, rather than making a production of pouring off bottles for use.
It is mandatory to get stainless steel spigots; the charm of the wooden spigots is quickly lost when they fail, leaking everywhere.
Does enough oxygen get in? Probably, but people who say so may be actively adding leftover wine all the time. It can't hurt to lift the lid whenever you think of it.
Allary oak vinegar barrels
Vinegar Shed (UK source)