Recently i installed my short shift base and handle for my T5, and ever since (i originally thought it was just cause it was uninsulated), i get vibrations fairly loud in the cabin

these vibrations only happen when the engine is on load, eg. full acceleration in fouth at 60kph. Yesterday i found that if i push down on the gear lever, it stops, as if im pushing the transmission away from the body.

was there a rubber mat or something that goes between the gearbox and body? im also getting heat transferred to the gear lever (possibly due to the friction in the vibrations)

There should be a rubber boot held on to the floorpan by a metal plate.

If the gearstick your running doesn't have that rubber insulator on it , I'd say that's your likely cause. Have a read of http://www.fordmods.com/forums/viewtopic.php?t=16741.

Otherwise, you could check if it's not the shifter itself - the floorpan hole might be large enough for you to get to the 4 bolts holding it on, check the lower shifter, and the plastic sleeve that the lower shifter ball stalk fits into.
As for heat, I'd still expect some heat to run through the shifter from the gearbox case.