Okay, so water is hindering for movement right?

So if my non-flyer starts movement in a square of normal hindering terrain and into a square of water terrain do they have to stop, or can they just continue thier movement at half speed?

Is an object that had to be dropped into water terrain considered water terrain or normal hindering terrain?

The second question stems from a situation where Aquaman got hit while carrying an object in water terrain and lost super strength. On his turn he still had his special power that lets him charge further from water terrain. But does he still get the bonus if he's starting his movement on an object instead of water?

They can continue moving at half speed. The only time a figure has to stop moving due to terrain is when moving from clear terrain onto hindering terrain.
Not sure, but my guess is that it's considered normal hindering terrain.
Well, that makes it even trickier. Again, I'm not sure, but I'd rule that Aquaman would be considered to occupy both hindering terrain (due to the object) and aquatic terrain (due to the map). In other words, he'd still get his Surge bonus, but would have to halve his movement since he started his move in hindering terrain.

Thank you.

I concur (if that makes a difference ).

how about Swingline?

does that let you run through/over water?

I believe it lets you ignor hindering terrain (for movement) and water is considered hindering terrain for movement.

Karduk is correct. A character with swingline ignores water terrain for movement purposes.