No.

We have a rough idea of which coins belong to Satoshi, based on when and how they were mined.

Satoshi's coins are continuously monitored and have never moved.

I always imagined that Satoshi probably lost the private keys to those early wallets.

Maybe. But in the latest Cra1g Wr1ght drama, he claimed to the court that some early addresses people thought might have been Satoshi's belonged to him. A message surfaced afterwards which was signed with those addresses calling Cra1g a liar.

The addresses used there were not Satoshi addresses, but other addresses mined around the same time.

I think it's more correctly stated that it's not public knowledge if they were mined by Satoshi or not. One can say they were not part of the "Satoshi miner pattern" that was identified years ago, but we don't know exactly how many of the blocks mined outside that pattern were or were not Satoshi's. Hence my "might".

Why would Craig (falsely) claim to own not-Satoshi addresses, instead of claiming Satoshi addresses that are less likely to be used (due to exposure risk)?

So the reason why Craig would do this is simple. Craig is not only a fraud, but is also an incompetent fraud.

And I'm confident that there's people dedicating a lot of processing power to try and determine its private keys (and anyone else's private keys if they happen to stumble across it).

I'm sure it's one of those tasks that'll cost a lot more than all the available energy in the universe kinda scale though.

It is possible but also in the early writings of Satoshi, he always stressed the importance of keeping your key and keeping it private.

What is the total value of his coins as of today ?

No. Satoshi's coins are scattered over hundreds or thousands addresses.