The trans-Pacific Partnership deal has shown a tremendous lack of support from both of the no-partisan Congress, and American history shows that you realistically only need the congressional majority party to support it to pass a bill. A trade agreement heavily agreeing with the Congressional majority party's values would almost definitively pass.
The United States Senate doesn't seem to get much of anything done these days due to such a stark line between the two parties. They seem to be happier blocking the opposing party from getting anything done rather than working together to get things done. Working together is what they need to start doing so we can get a trade deal on the table that works for both parties.
Yes, it is very possible to make a trade agreement that will receive congressional support. The trick is to get a congress that will work together. Right now too many congress members do everything in their power to oppose each other. If they would instead cooperate, then good trade agreements could pass.
Yes, it's theoretically possible. However in this political climate, with this administration constantly being stonewalled by congress, I don't see any way a trade agreement not drafted by the GOP makes it through. With all the stalemates and shutdowns and infighting, I just don't see it happening. Nothing is really getting done anymore.