Note that the  title  "Third   Foundation" was invented     completely

independently from  David Brin's  not  so soon  to  be released  Third
Foundation, an upcoming  sequel to the  Foundation universe.<<

I was wondering if you had heard of it. I'm a fan and have every book, if you need someone to verify the Foundation aspects of your story, I'll be glad to help.

The sharp ones out there will probably realize how I came up with it.

Lemme guess, another group to monitor the progress of Seldon's Plan? If this is the whole story, it isn't very appropriate.
Somehow, I think you ought to change the title. Avoid confusion with the book.
BTW, have you read "Foundation's Fear" by Gregory Benford? If you haven't, wait for it to come out in paperback before getting it. Not worth the hardcover.
I'm beginning to feel tired of hearing about Hari Seldon. I hope the next 2 books take place after "Foundation and Earth". It's a much more interesting time.

(really, really ancient, like before  spaceflight was  invented)

Not THAT ancient, unless Yakumo was around before 1957 and I recall he wasn't. But what's a couple of hundered years compared to 20,000?

"Hello, I am Hari Seldon," Seldon said after approaching him.

For some reason, I have this image of his hologram appearing in the Time Vault in my head....

(who was probably also a robot like Demerzel, though  she had never  >admitted it openly)

The way Asimov wrote Seldon, he apparently never suspected that she was until the end.

He could hardly  remembered how he  first met Daneel.  It seemed to be

somewhere in one of  the underground cities on  Earth just before  the
planet started  to become  intolerably radioactive.<<

By that time, there would have been no underground cities left.

If he had become human, he  wouldn't have grown so old as to need  one.

Yup, would have needed so many brain cells for memory, he wouldn't have had any space left for thought. 

He needed to go to Mycogen next, and that place was depressing enough as it >was.

Does he? Mycogen food is probably sold in various sectors throughout Trantor anyway. And does he live on Trantor or offworld?

except   that the    food there was

incrediblly good that it would disappoint Pavarti greatly if he didn't
get  some for  her.<<

Speaking of Pai. How is her memory?

 Her love  for  good food   certainly hadn't ebbed through the years.

Under the circumstances, you should use 'millenia' instead. It would sound better.

Really look forward to seeing the rest of this, if there is any...