I have no idea what those connectors are called. Your local metal fabricator will be able to knock some up though. I suspect a cheaper approach would be to bolt the horizontal beams to the vertical columns (but that won't work if this needs to be dismountable).
Your timber frame house will have timbers in the walls that you can anchor things to (although ...
If you're going to build a wall, just build a wall.
Framing lumber is pretty cheap. The only trick being that it can be a challenge to find straight pieces.
Get your measurements for the long section.
Build the long section on the floor (2" x 4", 16" on center is perfectly sufficient as Ashlar mentioned in the comments).
Attach the built frame to ...
Will this be stable enough as far as warp is concerned?
The term 'warp' can cover a lot of issues. Your boards may readily cup, twist, bow, or crook due to the repeated exposure to water and uneven drying.
The amount and type of warping will depend largely on the arrangement of the grain as much as how the boards attach. In general a short stub tenon ...
You've got hydraulic jacks - even cheap ones will lift 2", so lift height isn't critical, so don't try to put your 4x4 vertically - that could be wobbly as it moves during the lift, and it concentrates all the lift force at one point on your structure.
Lay 2 4x4s down sideways so that they cross all the 2x12s near each end.
Leave clearance for the ...
If you use pocket screws for all the major joints here you don't have to have any particular worries about strength, they are very very strong if everything is done right1.
Since this will see exposure to some weather (and the water from watering the plants I guess) be sure to use exterior-rated screws in the pocket holes.
How do I prevent it from tipping ...
The fact that your main post is flexing under load isn’t surprising given how far out the horizontal beam extends. On the up side it tells me the joint between the two is very strong, so, well done with that.
Adding an additional post to your main post (to the side facing or opposite your hammock) would definitely make the design stronger and may be ...
In the British tradition, such work would be done with a wooden bow saw (also called a buck saw). A nice feature of such a saw is the direction of the cut is nearly independent of where the handle is oriented, so you can cut a deep tenon as deep as you like, assuming the saw's frame is free to move. Such saws are in Europe in place of backed saw.