No. Prior to the creation of the country, some extreme proponents of Israeli independence were terrorists (the Irgun, among others, if I remember). But as a state they simply reply to force with force, a matter of survival when surrounded by governments representing tens of millions of people who vow to exterminate you. I don't always agree with their use of violence, but they don't hijack airplanes full of innocents, kick old men in wheelchairs off cruise ships, murder civilians for the sake of making a political statement, etc.
One man's terrorist is another man's freedom fighter. By Palestinian standards they're terrorists and by Israeli standards the Palestinians are the terrorists. About all anyone can say for sure is the entire region has an eons long history of violence and the Arabs are more likely to attack each other than outsiders. Palestinians being an exception to the rule.
no, they are the only democratic country in the region, surrounded by a bunch of kooks
They do support secret missions to kill enemies that would probably be considered by those people as terror tactics. Most militaristic states do that, though. It's always about perspective. Terrorism is just part of war. This idea that only the "evil " try to scare their enemies into submission isn't realistic.
No. Islam is a terrorism state. Israel is surrounded by terrorism.
Islam isn't a state, it's a religion, and only a very small proportion of its adherents engage or encourage terrorism.