Obviously while freely moving the object has...seemingly no angular momentum (isn't rotating)
Here is one error. If you pick a point along the straight line path of the object, then the angular momentum of the object is $0$. But if you pick your reference point as, say Jupiter itself, then there is a non-zero angular momentum about this reference point (assuming the object isn't heading right towards jupiter.
This is because the definition of angular momentum is $$\mathbf L=\mathbf r\times\mathbf p$$
Since in the case of using Jupiter as the reference point $\mathbf r\times\mathbf p\neq0$ (these vectors do not point in the same direction), we have a non-zero angular momentum. Furthermore, before having substantial influence from Jupiter, this angular momentum is constant because no net torque acts on our object.
Firstly object enters orbit around Jupiter. It now seems to have angular momentum and maybe potential energy as well? But no linear momentum? Is this all correct and what did the gravitational field do to cause the changes in energy?
The object keeps the angular momentum it had when coming into the orbit.$^*$ The object still has a non-zero velocity, so it still has linear momentum, as $\mathbf p=m\mathbf v$. The gravitational field does work on the object. Since gravity it conservative, we can easily consider the energy in terms of gravitational potential energy. As the gravitational potential energy of the object decreases, its kinetic energy (speed) will increase. If gravity is the only force acting on our object, total mechanical energy will be conserved the entire time.
It seems like you are thinking "Moving in a straight line $\to$ linear momentum. Moving around a curved path $\to$ angular momentum." This is not the case, as the above shows.
Secondly the object doesn't enter orbit but is partially deflected. As it moves in a curve for a short time does it have angular momentum at that time which it then loses (or possibly keeps by gaining rotation).
Since gravity is a central force, if this is the only force, and if your reference point is the center of this force, then there is no net torque acting on the object. Hence it's angular momentum will be conserved.
$^*$If the object "came from infinity" then this will be a hyperbolic orbit, not an elliptical one around Juptier. Unless there is something taking energy from the object