At the very start I got arrested and was to be executed for the crime of crossing the border. But skyrim is a protectorate of cyrodiil, isnt it? Even with rebels in skyrim, why would it be illegal to cross the border?

It's not because you were crossing, it's because you stumbled into the Stormcloak ambush and were assumed to be an ally of theirs. Hence, treason.

You were arrested with a bunch of stormcloaks. I believe that you were mistaken as one and that's why you get executed.

You weren't arrested for crossing the border in as much you were arrested for being in the same area as a bunch of Stormcloaks. You were crossing the border close by, immediate suspicion is that you're up to no good with the Rebels.

Civil war plot armor.

Wrong time, wrong place.