I'd recommend you also post the scramble you are solving, that way it is easier to give suggestions, especially for the cross.
Solving the cross one edge at a time is very slow. Solving them into the slot where they are supposed to end up is even slower. Remember that you can move around the D layer to bring the slot to the piece, instead of always bringing the piece to the slot. For example, when you had red and orange solved, you had both green and blue badly oriented at the bottom. They are supposed to go opposite to each other. At 1:41 in the video, with green cross edge facing towards you, you could have lifted both green and blue to opposite sides with F and B, then do a D move to put bottom layer in correct position relative to blue and green, insert blue and green with R and L, then D' to align all edges correctly with the centers.
Here's how I might have solved that cross:
Yellow top, green front:
B R F L F' D' R2
Breaking it down:
B R - solves green cross edge on orange face and moves orange cross piece out of the F layer
F L - solves blue cross edge opposite to green cross piece
F' - solves red cross edge into correct slot in relation to green and blue
D' - moves green, red and blue into their correct positions
R2 - solves orange cross edge
First F2L pair, you don't need to bring the slot to the piece like that. With orange facing towards you, you can align the pieces correctly in relation to each other immediately with R' U R (like you do later), then do a U2 to move corner to the slot where it is supposed to go and insert into back left with L U L'. Inserting into the back slots should always be preferred, because that helps look ahead. The solved corner is immediately out of the way at the back and you have more unsolved pieces in front of you instead of at the back of the cube.
Second F2L pair, note that you could pair them up immediately with a U move. Then L' U L to move them together out of the L layer, rotate cube 90 degrees clockwise and again insert into back slot with L U2 L'.
3rd F2L pair, 5:30, it is no problem to hide the corner piece when taking out the edge. Any piece you hide will be brought back to the U layer when you fix the cross. R' U' R hides corner, takes out edge, then brings corner back to U layer.
5:54, you have green and orange already paired up, could be inserted immediately with L' U L. Obviously this changes the blue orange case, but that's no problem. It actually sets up another easy 2 gen case, R U2 R' U2 R U' R'. Later, at 6:00, when you have paired up orange and blue, green and orange are in correct position to immediately be solved with L' U' L, without breaking up the blue orange corner. This is basically look what look ahead is about. Once you get used to performing the moves you are doing, you should be looking for the last pair, in this case green/orange, while solving blue/orange. Then you can notice shortcuts like this. But no worries, I know it can take a while to get there.
Edit: I should add that you have clearly understood the instructions correctly and are doing everything right according to the introduction videos. No problem there.

My comments were mainly about how to take it to the next level.