About the flex line from the washing machine and the line from the sink are they on the wall, in the wall, or under the floor?
Your question 1. You need to figure this out. Perhaps get a plumber who has a drain pipe camera.
Your question 2. Since rainwater and ground water are draining OK you should not need to do anything here.
Also I would ...
3. Reroute the downspouts to send the water over ground and away from the house.
4. Have the washing machine drain elsewhere, not into the basement drain system.
5. Snake the floor drain that is now plugged up.
The system (perimeter drain; weeping tiles; French drain) and any appurtenant underground channels to dissipate the water keeping the basement dry should not be inundated with water from other sources such as roof or washing machine.
If you determine for a fact that there is a tunnel, which is now functioning not as a sump but rather as a drain of seemingly unlimited capacity, then you can go ahead and use it for disposal of rain water and gray water.
Caution: Do not attach an extension hose or pipe to a washing machine drain hose. If the hose that came with the machine won't reach a place where waste water can drain by gravity, you will need a tank with ejector pump. The washing machine's own drain pump doesn't have the strength to perform that task.
When cleaning anything including baby diapers, it is better to sweep off or scrape off as much solid and semisolid material as you can first. (1) Less cleaning fluid or detergent needs to be used, (2) Less of a mushy mess which in turn means that bleaches or disinfectants will be more effective.