Could a TPA receiving a check on behalf of a retirement plan participant from a custodian or trustee deposit it in the TPA's account, with the TPA then handling the payment process and actually writing a check to the participant? What about the TPA being given checkwriting authority by a trustee over a trust account?

I do not see why not. The bank I used to work for was both a Trustee for plans and the paying agent for plans it did not Trustee.

see recently published IRS Announcement 2002-12, and IRS Reg 1.408-2(e) re custodial accounts, nonbank trustees, nonbank custodians. My opinion (also stated in the Announcement, 7th paragragh) is that once the money leaves the "qualified" trust and goes into any custodial account that does not have the protection given under 1.408-2(e), it is no longer qualified money.

Suppose the TPA is an individual. Don't you have a "plan asset" problem? Wouldn't the amounts have to be held in trust? Would the plan document have to appoint the TPA as the payer? Wouldn't the TPA have to be bonded?

Is this something that is done regularly?

Why not do it the right way? Why would you want to do this in a manner that would create doubt about exactly what transaction took place? Why create a "muddy" situation that you have to explain to everybody every time?