I would bring the bike inside at night. Even if it means bringing it in when the last person goes to bed, and then putting it back outside first thing in the morning when the first person wakes up, and leaving it in the middle of the living room while everyone's asleep and not using that space. The nighttime dew is bad for bikes, worse than daytime rain for those of us living near the ocean. The rain actually cleans the salt out of the air, whereas the nighttime dew combines with the salty air to speed up corrosion.
I would reach out to the manufacturer about what they would recommend for where the battery goes when the battery is not present. I'd hate to recommend something that's not congruent with what the manufacturer recommends.
I'd replace the rusty u-lock if it were me, and go with either a new u-lock or a strong chain lock. If your bike has standard 9.5mm drop-out quick-release levers, I would replace those with bolt-on or locking skewers. That way your wheels are either bolted-on or locked onto the bike, rather than easy-steal. Alternately you can carry two locks around (one to lock up both the frame and the rear wheel, and one to lock the front wheel to the frame). Your rusty old u-lock could be the second lock in this scenario.
This advice is based on what's sufficient to fight most bike theft where I live. I recommend that you ask bike shops near where you live as they have on-the-ground info that I don't have!
