If you are talking about the same structure at 22:40, no that is not a valid structure because there aren't enough carbons. The main carbon chain has 6 carbons and 1 carbon sticking out (a substituent), so there is a total of 7 carbons in the structure. 

Same with your lewis structure it is missing a carbon. 

You are able to place the CH3 on the top and bottom like you have in the lewis structure, that's the same. Just make sure to count your carbons so you don't miss any. 

It all depends on the carbon it is placed at, this goes back to 11:30. See below image explanation. 

You also are able to number the chair how you would like of course I recommend choosing whatever makes sense for the bond you have. This is why at 29:52 I labelled Carbon 1 in this position because I wanted the bulky group of isopropyl to be in the equatorial position and be pointing up. 

Noa Haviv This specific reaction is shown in the below video at timestamp 13;31 

In Organic Chemistry you'll notice that the lone pairs won't always be drawn in for you. But they are still there, just like the Hydrogens are not always drawn but they are still there.

Both nitrogens do have lone pairs, this goes back to the typical lewis structures and valence electrons that nitrogen has which is 5. If you need a refresher on bonding I highly recommend watching the below video. You'll see the common bonding patterns for nitrogen at timestamp 43;32 of the below video. Also in the below video I cover formal charges (super important) at timestamp 1;05;03.

The Cl-Cl bond comes from Cl2 this is how it's bonded together. At the very start of the video you'll notice the chemical reaction with Cl2. At timestamp 7;46 of this video, in the pink box this is what your question can look like, what is given to you is the Cl2. It is assumed that you know the lewis structure of Cl2 is Cl-Cl.