We are having this problem with a new house that is only 2 years old. My dad already ran the snake through the drain and a bunch of goodies from the toilet came out... But the problem still persists.
Please help!

That is because of two things, the basement drain is your lowest plumbing fixture and second you have a partially clogged main after that drain location. If it is a standard floor drain you can rod it after removing the cleanout plug. How to Unclog a Basement Floor Drain | DoItYourself.com

Thank you for your help!!

So we need to continue to use the snake until the drain thoroughly cleared? Ours is about 6'... but I'm guessing this is not long enough to do the job?
My father asked if it's possible that this could also be caused by the sewer outside? It is located right in front of our house. Also, is it normal that rocks and stones are caught by the drain?

Thank you again!

Thank you for your help!!

So we need to continue to use the snake until the drain thoroughly cleared? Ours is about 6'... but I'm guessing this is not long enough to do the job?
My father asked if it's possible that this could also be caused by the sewer outside? It is located right in front of our house. Also, is it normal that rocks and stones are caught by the drain?

Thank you again!

Chances are that you are going to need a snake a lot longer than 6 foot. Rocks and stones indicate a broken sewer line.
If the house is only two years old, you definitely should have a clean out on the outside of the house, between the house and the street.

In many cases a snake must be up to 200' to reach from the floor drain out to the city street riser at the sewer main. I'd consider getting a pro to give you a bid