Connect downspout to PVC pipe

This page says clay soil absorbs water at the rate of 1/2 inch per hour, 4 times slower than sandy soil or loam.

http://www.ornamentaledibles.com/tips/soil.htm

Let's assume you get 5-10 gallons per minute or 300-600 gallons per hour. A gallon is 231 cubic inches or 462 square inches 1/2 inch high. So 600 gallons/hour needs 277200 square inches or 1925 square feet. Assuming your trench is 12 inches wide, you need 1925 linear feet of pipe under steady state conditions.

Let's consider another situation where you store all the rain water and let it absorb as slowly as it wants. Assume you get 600 gallons which requires 80 cubic feet. So if your trench were 12" by 12" you would need 80 linear feet of pipe.

For the second question you might apply some septic dye into the old pipes. But often things get into the basement any old way, and then get trapped in the sump pit, rather than entering via the sump pit.

If you have a cleanout you can look into, septic dye might also address the assumption that they connect to the sanitary sewer line at all. Perfectly common (but also not to be assumed without checking) that they just go out to drywells in the yard.

If they do connect to the sanitary sewer, you need to be careful not to plug the sanitary sewer if you are plugging the pipes. Inserting an expandable "test plug" to seal the pipe is a reversible means by itself, or can be a way to make sure that concrete does not go down the pipe too far and plug up what it should not. A balloon is a cheaper method for the latter, though one should put only a small quantity (2" or so) of concrete on it at first, and apply more concrete after the first section has hardened, so that excessive force is not applied to the balloon.