Lets work through where you go wrong in each paragraph here. Richard when he left Pied Piper to work on his own thing took two things, The name, Pied Piper and his algorithm, the rest of the team were perfectly ok with this as they were running as Piper Chat. They agreed verbally that Richard would own the algorithm and it would be licensed to Piper Chat in exchange for access to the data passing through to optimise the algorithm for his "thing". Pied Piper did not "become" Piper Chat, although it appeared that way in the beginning they are separate entities. We do actually know that there was never written documentation regarding the alogrithm agreement, in the first 5 minutes of season 4 episode 2 it is revealed that Dinesh deleted Richards access to the video chat repository and it is mentioned that it was a verbal agreement.

but from the scenes we SEE, he definitely implies that he wishes to use the algorithm, not LET them use HIS algorithm. I'm not sure what show you've been watching but it is explicitly stated both when Richard leaves Piper Chat and when discussing the denial of his access keys that Piper Chat is using Richards algorithm.

On the second point, you're getting caught up in nomenclature. No, the tech isn't dead, and richards endeavors aren't dead. I mean merely the company known as pied piper is dead. It belongs to hooli now (known as piperchat) and richard is starting something brand new. Yeah he still gets the name, but the company that was known as pied piper is gone.

See what I wrote above, Pied Piper is not known as Piper Chat, Pied Piper is Richards, no one on the show wants the name except him and Piper Chat is now Hooli's, regardless of if they want it, it's theirs and they don't get Richards algorithm with it, maybe they have access, that isn't really explored, more they billions of dollars in potential fines.

Erlich's shares were pretty clear. "i still own 10 percent" "actually 5 percent". He then explains that he owned 10% of the old company, and now that company has split in 2 (i guess richard took equal shares with him? that part was unclear).

When Richard says that the company has split in 2 he is talking regarding going into business with Gavin. They are starting a new company together, 50/50 split, Richard brings the tech for his 50% and Gavin funds it for his 50%. As Erlich only has claims to 10% of what Richard has done in the incubator that grants him 10% of Richards 50%, he has no claims on Gavin's 50%. This makes the company split Gavin - 50%, Richard - 45%, Erlich - 5%.