Yes. It may be hard, but it is possible nonetheless.

It depends what you mean.

In the unmodded game, you can destroy battleships in the storyline (like the Unity in Mission 4), however you cannot destory Battleships like the Missouri near Zone 21, because they are bases. Basically, if it has a health bar, you can kill it.

Note that some battleships and cruisers in SP count as SOLARS rather than SHIPS, so you will not get any credit for taking them down (even though you can destroy them).

basicaly no you cant apart from a very few ones in the storyline
if you klick on it and a "health" bar comes up its killable otherwise you can do nothing more than temporarily disable a turret
if its dockable it isnt killable is the main rule to remember in the storyline

Yeah once again it is late but it is entirely possible i normally like takin them down with a sunslayer torpedo, just do torpedo runs on them while firing you guns it doesn't take long, in SP you haev to kill some for the storyline, part where ur in Sigma-13 and the rheiland fleet is there, i forgot about the storyline for a whil and just killed battleships till i ran out of torpedos

One mans Terrorist is another mans Freedom Fighter