Concrete – Unfinished basement – options for exposed concrete wall in finished room

I would definitely be concerned, mostly about mold but also about property damage (you said it's used for storage) and long-term foundation damage.

If your downspouts are not properly discharging water away from your house then you definitely want to address this. In the grand scheme of wet basement solutions, this is probably the cheapest fix you will find. Tight property lines or not, make sure that your downspouts discharge water far away from your foundation. It's hard to say whether you would benefit from a dry well.

If this fails then you should investigate whether you have a French drain and/or weeping tiles and whether that system has failed. If it is just clogged it could be a somewhat easy repair. If you don't have one or it has failed beyond repair then you will need to install or replace the drainage system. You said your house has tight property lines so it might not be realistic to dig up the outside, in which case you'd have to look into an interior French drain and sump pit solution. Either way this is probably going to be expensive.

I would definitely advise you to bring an expert in to review your situation and suggest some solutions. Make sure to get a few opinions/quotes. Fixing this this is probably outside of the DIY-scope for most people as messing with your foundation is risky business.

Yes, you can. However, doing it properly requires solid knowledge of how to transfer structural loads. The soundest practice here is to hire a structural engineer to draft plans for you to tell you exactly what you need to build to keep your house safe.