A/C that sends condensation into dirt?

Hi, hopefully I'm doing this correctly. Please forgive me if I'm not.

I'll start by saying, we have lived in this house almost 13 years. For the first 3 years there was no a/c as house had been built in early 70s. We put in a new furnace and a/c and mostly it works fine except for 2 issues that I'm now trying to fix.

1. I've slowly started to take over the household chores from my husband, and realize now that every spring/summer when the a/c is on, that there is a hose with water that comes out in the basement portion? I have no technical terms, but from what I'm reading this is like a hose, that is supposed to go into a floor drain. The problem is it doesn't. There is just a roughly half inch hole in my cement floor and it awkwardly runs into that but its too much water so we use a bucket that had to be emptied morning and night. This seems wrong.

On the other side of the basement room is a floor drain and an empty sump pump hole. Should the hose be run over there? Do I need a condensate pump (which I've just read about)? Is this something the original installers of the a/c should have done and both my husband and I were too dumb to know? And is there a ballpark price for what this would cost to fix in southern Ontario?

2. The a/c works well in most parts of the house (its not a large house by any means) but it doesn't seem to have any force in the main bedroom. The room directly beside which is also the furthest point is cold. Our bedroom needs 3 fans going to even get it somewhat cool.

Again, we assumed there was something in the ducts but had them cleaned years ago, and no difference. Do I need to take my ceiling down in the basement to open up the duct that leads to this room? Is there any other fix that might help before I go to that step?

Thank you for any and all help. I appreciate it.