Well yes - they were on one side, the Persians on the other.
Thermopylae was a very minor event in the 50-year Persian Wars. It failed to have any influence on the wars, and only later became symbolic of courage.
yes the Greeks went in a few wars
No, they already had a couple of thousand, one for each aspect of life, and didn't need or think of any more.
Yes, such as the Persian Gulf War.
King Leonidas is remembered for his role in the Second Persian War. As Sparta did not participate in the First Persian War ten years earlier, it is most likely that any other fighting Leonidas did would have been against other Greeks.
Yes. In modern times, Greece fought in World War 2 and Korea. In ancient times, Greece fought the Persian Wars.
Yes, many. The Greeks fought wars constantly with each other and with outsiders. The Trojan War happened so long ago that it is difficult to glean truth from myth, but the two Persian Wars are fairly well documented, and there is a very good account of the Peloponnesian War between Athens and Sparta. The dreaded Philistines of the Old Testament (Judges and Kings I) may have been expatriate Greeks. The Greeks continued to fight many wars with the Persians whom they finally subdued, and later with the Romans who finally subdued them.
Governments conduct wars. They've got the budgets for them.
No more than all Europeans are Greeks, or all Americans are Mexicans. Asia is a vast continent with countless cultures, ethnicities, and languages. It would be ridiculous to call a Persian or Indian person Chinese, as they are not even ethnicially related. (Non-Chinese) East Asian and South East Asian cultures are also quite different from one another. The reason for the similarities are due to China's influence on them. However, this does not make them any more Chinese than the influence of the Greeks and Romans on Britiain makes British people Greek or Italian.
Alexander the Great spent about six months in Egypt. This was part of his campaign against the Persian Emperor, Darius III.
At the time of the Peloponnesian War (431-404 BCE) Israel was under control of the Persian Empire, and so in no position to intervene in any of the Greek wars, not that it would have had any reason to meddle in Greek affairs anyway.
Ancient Greece did not expand into the Persian Empire. Greece was comprised of over 2,000 independent city-states, the ones in Asia Minor being inside the Persian Empire. It was the Macedonians under Alexander the Great who expanded into and took over the Persian Empire. Alexander's successors divided the Empire into kingdoms of their own (Egypt, Syria-Mesopotamia etc , and these lasted a couple of hundred years until the expanding Roman Empire absorbed them.