Carr wanted to be traded prior to restructure per NOF and CBSsports

There was no world where Derek Carr took a paycut to be traded.
In general it's unlikely that any player will do that, not just Carr. It happens on occasion but its not common. In that I don't blame him. If it's your last chance at a big payday anyone would do the same.
1) He has signed his last multi year starting QB contract. He wouldnt give that up.
NFL= Not For Long, yet he has somehow manged to buck that trend. Agreement though. He is next likely to be a bridge or a clipboard holder. It's possible he is already a bridge QB now. That he has avoided that so far speaks to the state of QB play in the NFL, not him.
The next deal he gets will be a 1 year deal, either to be a bridge to a rookie QB (which is hopefully what he does here for 6 weeks) or maybe to become the new Vinny Testeverde/Joe Flacco where teams call you when the rest of their QB room is injured.
Agreed to a degree, more apt description is Ryan Tannehill though.
2) You cant trade what no one wants.
If the NFL is led to believe what happened last time it was down to the Saints and the Jets?

No way I'm buying that, not even for a dollar (queue the reference). The Saints were in a "bidding war" of one: against themselves and they "won."

I suppose its hard to lose when you are only playing against yourself as it were?