Use of transformer in arc welding
Modern welders usually don't use mains frequency transformers, rather they are sophisticated IGBT or MOSFET-based converters from mains to DC or synthesized AC (AC is required for welding aluminum, DC is used for steel). 

When mains frequency transformer were used (and simulated with modern inverter type power supplies) there are really two types- more-or-less constant voltage (CV) and more-or-less constant current (CC) type. The latter is achieved by the use of magnetic shunts in the transformer. CC is used for TIG (Tungsten Inert Gas) and stick welding. CV is used for MIG flux core welding. 

The purpose of the transformer (or switching supply) is to create the CC condition for TIG, to reduce the voltage and increase the current, and to provide some galvanic isolation from the mains, and for big ones, to convert 3-phase input to the required output. 

As far as what would happen if you touch the torch- the voltages used for the arc are usually well under 100V, so it's not particularly dangerous, however most TIG welders have a HF (High Frequency) start circuit that generates some thousands of volts at some mA to get the arc going (the ones without HF are called scratch start). It would not be pleasant to touch that. In the old days this was done with a spark gap, kind of a Tesla coil in the welder, inductively coupled to the torch welding cable. 

If you're welding you're going to have leather gloves and have the work properly grounded, so it's not so common to get a shock from the HF but it happens.