You don’t take standing loss for shooting other players.
Enemy militia are legal targets anywhere and there will be no sec status hits for shooting them. You will encounter neutrals in low sec however and all the normal rules apply (you only lose sec status if you shoot first AND they are NOT wartarget/suspect/criminal/outlaw/limited Engagement)
Joining faction warfare allows you to travel in friendly faction space as normal, but you are a WARTARGET to members of enemy militia which means they can come after you and shoot you even when you stay in friendly space.
So if you are a member of caldari militia you can still be killed in caldari hi-sec by gallente militia. This includes when running a hi-sec mission.
For this reason using slow non-pvp ships, even in hi-sec, is somewhat risky. Typically FW players will not do much transportation unless it’s in fast/cloaky haulers and they will rarely run missions in hi-sec.
With regard to your characters, if you still want to run hi-sec missions and/or have free access to all of hisec with your main you should not enter that character into FW. Which means he may need to learn to haul for himself if you move large items.