Conversion of angular momentum to linear momentum in free space If two objects both with angular and linear velocity collide in free space, can the total linear velocity of the objects increase at the expense of a loss in angular momentum?
In other words, imagine we have two dumbbells thrown at each other. Both are moving forward and spinning. 
When they collide will some of the angular momentum be converted into linear momentum or will linear and angular momentum be conserved independently?
 A: Conservation of linear momentum and conservation of angular momentum both follow from Noether's theorem, which shows how conservation laws arise from symmetries.
Conservation of linear momentum arises from translational invariance - that is, if you shift space in some direction, the laws of physics are the same. Conservation of angular momentum arises from rotational invariance - that is, if you rotate space in some direction, the laws of physics are the same.
Consider a particle moving in one-dimensional free space. The Lagrangian is
$$L(x,\dot{x},t)=\frac{1}{2}m\dot{x}^2$$
Taking the transformation
$$t'=t,\quad x'=x+\epsilon,\quad \dot{x}'=\dot{x}$$
This then gives
$$\delta x=\epsilon,\quad\delta\dot{x}=0$$
Noether's theorem states that there is a conserved quantity for some generalized coordinate $q$, which is
$$J=\frac{\partial L}{\partial \dot{q}}\delta q-F$$
That is,
$$\frac{\mathrm{d}J}{\mathrm{d}t}=0$$
Here, it is simple to see that $q=x$, $\dot{q}=\dot{x}$. Therefore,
$$J=m\dot{x}\epsilon-F$$
From here, it is easy to see that $m\dot{x}$ is a constant, that is, $p_x$ is conserved.
We can construct a similar Lagrangian for any number of bodies in any number of dimensions, and find that the total momentum of the system is conserved. Indeed, we can also do this for angular momentum. If we do so, we find that these two conservation laws are independent of one another. Any momenta is conserved, regardless of the other. Therefore, both are conserved, and neither can be "converted" to the other.
A: I would like to give a different argument:
Linear momentum and angular momentum have different dimensions - they are essentially different - you can't get apple from orange, nor can you add apple to orange.
Since the dimension of angular momentum is equal to dimension of linear momentum times length: 
$[J]=[P]\cdot L$
Obviously, there is no way to equal $[J]$ to $[P]$, so the answer to your question is: Nope. 

The flaw in your argument is: in fact, there is no increase nor decrease of angular momentum afterward. If you sum up the changes of angular momentum of two tops after the collide, they will result in zero, as angular momentum is conserved in your thought experiment. The same for linear momentum.
A: I do not know mathematical proof but No, that should not happen. 
Angular momentum can be a spinning body, or rotating body.
If the spinning body hits another body, the bodies may bounce from one another, but that would not change the linear momentum of the system, because, the bounce will be in opposite directions and linear momentum of system will remain unchanged.
If rotating body hits another body. Rotating means one body is orbiting another body, or both the bodies are rotating around one another. In this case, also, the angular momentum is due to the mutual attraction of bodies. And when they collide, they equally push/pull one another to merge.
The question then would be, where the kinetic energy due to angular momentum goes - That is lost in the collision, or converted into rotation of the combined body.
A: The total angular momentum is conserved. But it is more than the spin of the objects.
Calculated from the origin of an inertial frame, each object has:
$$\mathbf L = \mathbf r_{CM} \times \mathbf p_{CM} + I \omega$$
where $I$ is the inertia matrix and $\omega$ is the column matrix angular velocity. $\mathbf r_{CM}$ is the position of the center of mass, and $\mathbf p_{CM}$ its momentum.
The sum of $\mathbf L_1$ and $\mathbf L_2$ is the same, before and after the collision. But magnitude of the spin component $|I\omega|$ can decrease (or increase) for both objects.
A: Yes, I just did a thought experiment that shows that indeed angular momentum will be converted to linear momentum.
Imagine we have two gyroscopes or tops spinning on a table. The tops each spin rapidly but move around the table slowly, i.e., they have a lot of angular momentum, but little linear momentum.
Now, imagine the two tops collide, what will happen is that both of the tops will shoot off at high speed, there angular momentum being converted to linear momentum.
It can be seen in this kenkogama video. When the tops touch they move away from each other much more rapidly than they approached.
