I’m not 100% clear on what you are trying to do here. Are you saying that you want a new email address that would be something like your_name@the_organisation_in_question.com? If so, it would be up to the organisation to set you up with an address on their domain; you can’t set up a Gmail account with a non Gmail domain name as far as I know.
However, what you can do is use the Gmail web client (or mobile apps) to access email from other services as well, as long as they have POP3 access. In fact, you can use the Gmail account you already have and add the new organisation’s email account to your inbox - and therefore manage all your email in one place. You’ll still need the organisation in question to set you up with an email address, though.
I’ll add that their web person set up a forwarding account, but it’s not working as either I get nothing or it go’s into my spam folder. I set a filter to keep it in my inbox, but it’s not always working.
Yes. They’ll even allow you to merge the mailboxes for the two accounts, which will probably be necessary anyway since otherwise you’ll have to login each time you want to look at the other mailbox.
You could use the “+” trick: Gmail will ignore anything between the “+” and the “@,” so if your current Gmail account is GaryM@gmail.com, give your business contacts the address GaryM+officeheld@gmail.com, then create a filter so anything addressed to GaryM+officeheld@gmail.com goes into a separate folder (well, label; Gmail doesn’t really have “folders”).
Alternatively, Gmail ignores dots, so you could give out the address Gary.M@gmail.com, then filter for that.