piston engines require a warm up period because the oil is too viscous to lubricate properly when cold with the "wet sump" arrangement that exists on such engines.
gas turbines are pressure lubricated, and the oil is synthetic and lower viscosity (as indeed some reciprocating engine oils are now...) and therefore lubricate effectively straight from cold.
no, gas turbines don't "overheat" with prolonged ground operation in hot climates. The limiting factor is the turbine inlet temperature, (i.e the maximum temp the turbine blades, disk etc. can cope with) and this is only potentially achieved in warm ambient air when the engine is at its maximum thrust setting. Idleing EGT will be a few hundred degrees lower!!