Q1: Assuming that by operating temperature you mean turbine inlet temperature, this article says the Pratt & Whitney F135 engine in the F-35 (JSF) has a TIT of 3600 °F (1982 °C / 2255 K), but that most high performance aircraft engines do not exceed 3000 °F (1649 °C / 1922 K), while non aviation engines are nearer 2700 °F (1482 °C / 1755 K) or lower. Exact figures for modern engines are very unlikely to be stated in public. Mattingly, in his book "Elements of Propulsion: Gas Turbines and Rockets" has a table (6-2, P363) that gives a general guideline, depending on the age of the engine. He says:
- 1955-1965 2000 °R = 1540 °F (838 °C / 1111 K)
- 1965-1985 2500 °R = 2040 °F (1116 °C / 1389 K)
- 1985-2005 3200 °R = 2740 °F (1504 °C / 1778 K)
- 2005-2025 3600 °R = 3140 °F (1726 °C / 2000 K)
The GE J79 gas turbine which had its first run in 1954 is claimed to have a TIT of 1710 °F (932 °C / 1205 K), so these guidelines look reasonable.
Q2: Increasing the turbine inlet temperature allows more fuel to be added, which increases thrust and hence thrust to weight ratio, and specific thrust (thrust per lb of air ingested). Increasing TIT (at the same pressure ratio) also has some effect on improving the specific fuel consumption. At the same TIT, SFC is improved by increasing the compressor pressure ratio. The chart below shows the effects for a particular engine, the specific effects will differ for each engine depending on the bypass ratio etc.

Q3: There are three methods that can be used to cool turbine parts. The primary approach is to bleed off a portion of the compressor exit air, which then bypasses the combustion, and is feed into the turbine vanes and blades. The below picture also includes some numbers for the % flow of compressor exit air that is used.

Other methods that are often used as well include the use of small holes over the blades and vane surfaces, through which the cooling flow emerges. This provides a film of cool air over the part, and is known as film cooling (left picture). On some more modern engines, thermal barrier coatings are also used (right picture). These are ceramic coatings that act as insultion of the base metal from the hot gas. Usually the TBC is added as an extra method to help gain additional part life. But sometimes, the TBC is needed to obtain the base life, and is then know as "prime reliant".
