First of all, when playing with new people make absolutely sure you know which rules and books are and aren't allowed. Psionics tend to be banned for various reasons, but the most popular ones are that it's too much work for the DM to understand and/or the DM thinks they're overpowered.
On to your actual questions though: If I understand you right, it seems you're under the impression that the claw attacks are free actions or something. They aren't. The claw attacks psywar usually gets are from the power
Claws of the Beast, which can scale up quite nicely actually. The issue with that power is it explicitly says the claw attacks can't be used if you're also using manufactured weapon or other natural attacks in the round. That's an exception to the usual rules that allow having multiple natural attacks on a full attack such as claws and a bite for example.
Besides the issue of not having those claw attacks work for a druid while Wildshaped, taking psywar levels will only slow down your druid progression. Druid is a stronger class than psywar, so only specific gimmick builds might benefit from the combo. Since you're a new player, I recommend you don't even bother multiclassing druid with psywar.
If you were thinking of using
Psionic Lion's Charge while wild shaped then keep in mind you'd need to be at least a 4th level psywar to manifest that power, which would cut into your druid levels significantly. There's no comparison between a psywar 4/druid 5 build and a druid 9: The druid 9 would win because it has spells much more powerful than the powers gained from psywar, plus he'd have access to stronger wild shape forms, and he'd have a stronger animal companion as well.
You are onto a nice idea for using the Synad race with either psywar or druid though. The
Rapidstrike and
Improved Rapidstrike feats are great additions to either.
