The first crash is caused by the Phase Jaunt ship system as zoe_zucchini pointed out, which means both of the ships that have that system (Abraxas and Nightseer) will crash the game any time they're fielded. Worse yet, both ships appear on the title screen. Luckily this is very easy to fix.
I don't know what causes the second crash. It happens in the fighter AI code of the core game, which is obfuscated, meaning I can't read it without a great deal of effort. If I knew the cause of this crash I'd probably be able to fix it easily, but figuring that out would involve watching hours of battles trying to identify a pattern.
If you're considering playing with ICE enabled I would recommend against it. Starsector is expected to update in a matter of days (but no promises), and the next version of ICE will break saves in order to improve the recall teleporter anyway.