I think it would be perfectly reasonable to record conversations with customer support. Whether or not it would make any difference to the end result is unclear. In most cases companies go to a great deal of trouble to make sure you don’t have a feasible route to take legal action against them. And often that’s the only real option available against a company. Often they don’t care about individual customers/clients.
Where or not it is legal to record them might or might not matter. It seems very unlikely to matter, since as the company is unlikely to take action against you. It might mean the difference wrt admissability in a court, but as I said above, companies are making it increasingly difficult for consumers to get there. I’m not sure if the EU, for example, is doing anything about this.
Oh, and if you can record customer service (or anyone else for that matter) I can see no reason to mention it, and obvious reasons not to. Few are going to respond positively to that disclosure. Certainly not employees of billion dollar corporations. It would be nice if companies were concerned with being fair. Unfortunately that’s not the world we live in.