How big would the L1 Earth/Moon or Earth/Sol zone of null gravity be?
A few 10s of km wide?
Last time I crunched the numbers, it was a few hundred or thousand kilometers across (between Earth and Luna). The Earth-Sol L1 jump point would be much larger.
I don't think Mars and Phobos would have a point of usable size.
Is their a danger of jumping onto another ship that’s recently arrived at the point?
If multiple military ships were making a coordinated jump, say, the Earth-moon L1 point, then it shouldn't be much of a problem. Errors in military ship jumps are about 500 meters. You can stack quite a few ships at 30-kilometer spacing in a jump point hundreds of kilometers across.
If two random ships were aiming at the same jump point and unaware of each other...well, standard practice (per StratOps) is to clear out of jump points - even standard points - right after arriving. JumpShips, space stations, and everything else move away so accidents don't happen.
So it's a pretty low chance of two ships landing atop each other at a pirate point.
I think the Jump Drive won't work if that point is wrong but it will still discharge the core meaning you have to recharge it again.
The jump drive - via the Brandt Recoil Effect - has an idea if the target jump point is safe. However, using a pirate point deep in a star system is inherently unsafe and there's a degree of forcing the jump. Most often, jump malfunctions result in a spent charge and failed jump. However, if the jump goes off during a rolled misjump then, yeah, the ship can be borked.