1) Did Jesus found a church?
2) If yes, can His Church teach false doctrine?
No. Church's are founded by men and built on faith. Faith is belief in what someone says based on some evidence. The degree to which what is said and taught is accurate regarding what God taught and is true in general determines how closely the adjective His fits when referring to any particular church. His refers not to something that is owned, but to faith in what He said and believes and cherishes, not to what someone else says.
"2) If yes, can His Church teach false doctrine?"
Any and all churches can teach falsehoods. If they are His, they will admit that and provide evidence instead of claims of infalibility.
He never said He was going to FOUND a church, He said He would BUILD it. There's a difference there.
2) If yes, can His Church teach false doctrine?
There is no entity of the church that can speak for itself; there are believers (or not) who can speak for the church, or who claim to speak for their local assembly, who can be wrong, but that is not the *Church* speaking.
The only things that speak to us are Scripture and the Holy Spirit.
Individuals can teach false doctrine, which is why Scripture is so critically important. It cannot change since it is written down. It's always the same and there to be appealed to if a disagreement arises. That's why IT needs to be the final authority vs. a clique of self-proclaimed mouthpieces for God.