What materials can be cut with the M A Series III press?

Most flexible materials like:-
Rubber  : Carpet & Textile : Felt : Sponge : Foam : Plastics : Cotton : Fabrics : Gasket Materials :
Polyester :  Textiles : Shoe Materials : Paper & Card : Cork : Leather : and lots more.
 

What thickness can be cut?

Quite a variety of thickness can be cut, but there are limitations.
(This is assuming the cutting die is in very good condition.)
Lets say for about a 200mm (8” approx) circle.
Some examples are :-
Thick Veg-tan Leather will cut fine even 8mm possibly 10mm. 
Very 'boardy' (hard or old) sole leather maximum is more likely 2 to3mm thickness.
Multi layer stiff paper perhaps less than 1mm, soft cardboard 4 t0 6mm.
Most foam 12 to 14mm .
Soft multi layer vinyl-cloth combination material 8mm.
Soft fabric 8 mm possibly 10mm.
Hard stiff cotton fabric 3mm.
Warning:-
Should this occurs, it is time to up-grade to a hydraulic machine, as you have forced the machine beyond it's drsign capablites.
As you can see the theme is all about the softness and flexibility, hard 'woody/boardy' product is somewhat limited, maybe small pieces can be cut.
While this information is for 19mm(3/4”) cutting dies, other die heights may be used, although a change of cutting board thickness is required (see spec. chart on M A Series III instruction folder).
While the head height has quite a range of adjustment height wise, operating clearance above cutting knife does not change regardless of die height or head adjustment, it is a little over 12mm (1/2”).
This last statement may be a little difficult to get your thought process around, for although you can adjust the clearance above the knife by quite a lot, the stroke is fixed so it will only cut correctly when adjusted to the correct point (about 12mm above any height die).