Hi,
You'll definitely have to deal with the physics engine I am afraid
because your requirements would simply mean writing a physics engine yourself...
now, yes physics can be difficult to get right, but I encourage you not to loose faith, and keep trying because it's only a matter of settings and tweaking.
1: this is easy, and not in contradiction with the engine, you simply switch physics settings to achieve this. for example, when the character need to float you attach it to a different rig, maybe a spring joint or simply switch off the gravity and increase the drag so that it drifts nicely.
2: do change the mass and i will affect the behavior under gravity.
4: to easily move a character, it really depedns what you are after, but setting a rigid body to "kinematic" will allow you to move it like any other non physics gameObject, else you can do like point 1, attached a sprring or joint and move the primar target and let the joint pull and push your character to follow that.
5: there is a system for this, explosion, I am about to release the shooting game from m2h and it uses explosive forces, so you'll have a working sample.
Now, to constraint all this into a 2d, there are ways, you can constraint one axis for each of your objects for example.
Have you worked with the various 2d frameworks available on the asset store, they are a lot of 2d games using physics, this would give you some working examples to learn from.
Bye,
Jean