What is going on with my downspouts?
If you want to repair the leaking pipe where it enters the foundation, I'd use hydraulic cement to seal it. As for the downspout into the cast iron pipe, there are transition pieces that should tie them together more cleanly and tightly - or you could fashion something from ABS or PVC sewer pipe/fittings and downspout fittings.  Otherwise, my thought on this one is: "If it ain't broke (or not causing any major problems), don't fix it"!

As DaveM stated, it appears that water from your downspouts is flowing into the sewer system. Although generally not allowed today, it was apparently legal when the house was built, so I'd leave it alone - unless you have flooding problems in your basement or crawlspace. Making any changes to this system would typically require you to follow current codes which could end up being very costly - or at the very least, time-consuming.