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So if I take a unit of thousand sons and I choose to use the psychic power my AS has can the rest of the squad still shoot?

Yes, only the Sorcerer is restricted by his Psychic Powers. You work out who can fire on a model by model basis, with the caveat that if one model moves all models in the unit count as moving.

I'm pretty sure they are all relentless with the combination of 'Sorceror Commands' and Slow and purposeful

Actually, everyone except the Sorcerer is relentless, as he doesn't have S&P as a special rule. But the caveat on moving was in a general sense, not specific to Thousand Sons who don't actually care about moving.

There is not a single psychic ability in the chaos codex that will ever prevent a squad of thousand sons from firing if their aspiring sorcerer uses it. However, i do believe theyre bound to shooting the same squad the sorcerer does (if its a shooting attack). Things like warptime and gift of chaos do notwant apply to the best of my knowledge. Theres a debate on whip, but i dont think aspiring sorcerors have it as an option. Wind of chaos, doombolt, and bolt of change all count as psychic shooting as they cause wounds and have definitive shooting weapon profiles.

Also not that since the marines in the unit count as slow and purposeful, they do have the relentless special rule, and movement does not effect the firing ability of their weapons (they count as stationary when firing for all intents and purposes).

Aipoch you mean 'Lash of Submission'?
I thought that was a slannash spell

1) Yes its slaanesh
2) No there is no debate. Lash is not a PSA according the BRB FAQ definition of PSA.

As i said, aspiring sorcerors do not have it as an option.

If its in the faq then its in the faq and needs no further debate. There are plenty of other threads discussing the proper mechanics around lash as it is.

thanks for the help, cheers!