How far should I insert downspout into sewer connection pipe?
In many places it's not legal to connect a downspout (storm water drainage from a roof) to a sanitary sewer system. The reason is easy to understand: if storm water flows into a sanitary sewer system the sewage treatment plant may easily be overwhelmed by rainfall. What happens when the sewage plant is overwhelmed? Untreated sewage escapes somewhere. Either the plant's output is not treated as it should have been or sewer plumbing upstream overflows.

On the other hand, if you're connecting a downspout to some kind of local storm water holding or drainage system, I suggest there should be a gap between the two near (above) ground level. If the underground drainage system should back up for any reason it's better for the storm water to spill out at ground level than for it to back up into the downspout and gutter system. By "gap" I mean use loose-fitting adapters so that backed up water can easily drain/spill out.

Gutters aren't always secured well; the weight of being filled to the top with water may tear them from the building. The backup may also encourage water to be wicked up into the roofing system. If the weather is cold enough for the standing water to freeze in the downspout and gutter, that brings a whole other raft of problems.