Searching for available Apple Developer account names I am about to set up a company but I have a suspicion another developer has taken the name that we would intend to trade under for a developer account.

We're setting up our business in the UK, and the official registration office (companies house) claims the name is available as far as 'the company' is concerned, but is there some way I can check if the Apple Developer account name will be available before actually forming the company? Will it be a problem or can we and this third party both trade under near-identical names? (for example fuzzy games and fuzzy-games).
As far as I know you can pick any name as you like, as long as your company details shared with Apple are correct. And yes, they check everything before you get approved - took me a few weeks to fix the details with them because they've missed the "Inc."

Do you really want to change your future company name if the account is taken as Apple ID? There are many many many companies around with similar names, or near similar names.

Just take the name you had in mind, as long as your company details match there is nothing they can do to disapprove your Apple ID.