Are there experiments that show mass actually increases as acceleration increases?

Answer 1

I guess you are referring to the apparent increase of mass (or inertia) in a moving frame of reference under Einstein's theory of relativity. If so, then yes, there are standard student experiments. Details of one such can be found here:

http://physics.dickinson.edu/~dept_web/activities/papers/relativity.pdf

Answer 2

Every particle accelerator in the world is engineered around special relativity. A machine that wouldn't take the increase of effective mass of particles into account simply wouldn't work at all. In linear accelerators you can see the resonators change in length along the path of the machine as the particles pick up speed and mass.

A proton cyclotron seizes to work for energies close to 1 GeV, the rest mass of the proton. Inventive people came up with a machine has specially shaped magnets and D-s to mitigate this effect and extend the machine energy:

http://ssf.ugent.be/linac/linac/tomography2.php?x=1

Looks a little bit like orange slices, doesn't it?

A microtron goes all the way and increases electron energy by a full rest mass per revolution:

http://en.wikipedia.org/wiki/Microtron
http://www.isa.au.dk/facilities/astrid/electrons/microtron.html

wouldn't work if not for the fact that Einstein got it right.

Your old color tv or CRT, by the way, was accelerating electrons to mildly relativistic speed. At 25kV acceleration voltage, the total energy of the electrons is 25keV higher than their rest mass of 511keV... a 5% difference. If you happen to be a CRT designer, you have to take this effect into account, or the tv set won't work right.

Good enough?

Answer 3

No, mass is a constant.

Weight on the other hand does increase with acceleration. since it is a measure of force that is the product of mass and acceleration.

Answer 4

Yes. An electron beam with a given velociyy deflects in transverse magnetic field of given strength by an amount inversely proportional to its inertial (relativistic) mass. You can measure it that way.