Are there restrictions on a dummy moving three hexes?
Nope. Per 5.1, "Forces and Dummies... may move at one of four speeds as the player chooses, but never faster than their slowest unit." What a Dummy's "slowest unit" is depends on what you're trying to emulate.
I also suppose you have to then look like you're fiddling with a fuel counter?
Unless you want your opponent to say, "You're not fooling anyone."
Can shadowing air sector assets be "shot down"?
No. Per 6.3.5, this opportunity should never arise, because "Shadowing must also end if the enemy force... enters a hex... adjacent to an airbase hex with Interceptors [1.1.2]."