Secondly, though not always the case, I might have to process their respective customers' data in some way in order to do my job (by having access to a production database, for example).
So, yes, GDPR absolutely does apply to my business.
No, I didn't say that. GDPR doesn't mean that you have to ask for consent in each and every case. If as a business you have a legitimate interest to store client data for a specific purpose you don't have to ask for explicit consent. Being able to contact clients in the future who contacted you first constitutes such a legitimate interest.
As for the right to be forgotten: Sure, why wouldn't I? If they don't want to be contacted anymore and want me to delete their contact info I'm happy to oblige.
Basically, it means that what's been best practice all along is made explicit. So, in most case this doesn't affect the actual workflow all that much.
However, in the past occasionally clients could be quite liberal in handing out access to their production data. GDPR definitely has raised awareness that this isn't something to be taken lightly.
It's been quite some effort but for my business so far it's been a net positive because it made me rethink, clarify and streamline my processes.