TPA has been passed by Congresses since the 1980's - despite what hysterical Trump supporters like to claim, there was nothing controversial about TPA. The issue is TPP, and Cruz is opposed to that, as are most of the GOP.
On the Corker bill, Cruz opposed cloture but was out voted. He explained his reason for voting for the bill on final passage:
"I voted no on cloture because we should have insisted on amendments to put real teeth in this bill. Ultimately, I voted yes on final passage because it may delay, slightly, President Obama’s ability to lift the Iran sanctions and it ensures we will have a Congressional debate on the merits of the Iran deal. I will continue to lead the fight to prevent the Islamic Republic of Iran from acquiring nuclear weapons and to protect the national security of America and our allies."
The subject of the amendment to increase H1B visas has been dealt with numerous times. That was part of an attempt to demonstrate that the Gang of 8 bill was about amnesty, not about border security. He showed that even if conservatives agreed to greatly increased legal immigration, the Gang still would not agree to a provision preventing illegals from being granted amnesty and citizenship.
By the way, Trump also wants to increase H1B visas, and he also plans on bringing back almost all of the illegals currently in the country almost immediately after deporting them. So it always amuses me when trump supporters try to use the H1B issue against Cruz...