For the lead change, I normally stop after the first barrel and pick up the lead again, then go through the trot after that. It makes me so thankful that Selena knows how to do a flying change, that made things so much easier with her...But It has been a nightmare with some others.
Now onto the videos...I am going to watch and then point things out as I see them :)
First thing I see on the barrels video is that he is not bent properly going into the turn. The first one wasn't bad, he wasn't fighting you and stayed on task, but if you can get some inside leg about thirty feet or so off and get him bending around it, getting soft, and looking at that barrel your turn will smooth out. This was the issue with the second barrel. He was looking the wrong way and you had to hold him off, which put him out of balance. The inside leg will soften him, bend him, and hold him balanced in the turn.
It's the same when you're loping, you have to get that shoulder with your leg and balance. Collect him, get him driving up from behind. He's kinda not putting any effort in right now, so you have to make him. The free-wheeling lope will make it all too easy in the beginning to have him bowing off the barrel. Get him softening and bending properly, it will make all the difference. I tend to underestimate that but every time that's what makes the biggest change between a mediocre run and a precise, on the money run. And if he bows off, circle it again or stop and haunch turn him to the inside. He should respect that he needs to finish that turn and left his shoulder out.
The next two times were better, but I would also emphasis pushing the hip in. Another thing I've noticed too is that there's no significant rate just loping through like that. You always have to rate, even if you can make the turn as a slow lope, always bring them back to you. If you're slow loping, trot around the barrel. My preference is to speed up between and then slow lope around the barrels. Otherwise they'll think rate is optional.
I think just those things, rate, collection, and bend will make a huge difference in Red :)
Now for the poles.
You had an awesome approach to that first pole, good distance, but still there was no bend. Bend to the turn is critical in everything, because it sets that horse up. It tells them, "Hey, get soft, get ready, we're going to turn soon." Because if you don't have that bend, they have to throw themselves into that turn and make twice the effort to get out of it than if they had the bend already. Horse can't make a good turn without looking at the pole or barrel at least once.
Also, for the first turn, I would take him past it two or three strides and then finish close. Inside leg, bend, trot past it, make a u-turn and finish close and straight. When you go fast it'll tighten up and make a sweeping turn motion. It you turn in a circle like you would a barrel, you'll bow off and end up not straight for the weave.
When you are actually weaving, put your inside hand up to the pole. It's more like a lifting motion than a turn. Think straight line, and lift with your hand just a little to give you room to get by the pole. Otherwise you have the horse steering in and out, which you may not hit anything, but you take excess steps and get slowed down.
And I'll definitely say it again...Collect that trot and soften his face.
I see your legs not making too much contact with him (I can't see very well but I saw it a couple places) go ahead and bump bump him up with your legs and hold him gently with your hands, some nice steady contact to ask him to relax, soften his face, and drive up from behind. You said he's heavy on the forehand and that will definitely help that out too.
When you loped the poles, and you got to that end pole, he was WAY too close both times. And you saw that, by telling him to get over. Your inside leg needs to be on him WAY before you hit that pole, as soon as you leave the one before it. It will hold him off of it. And just like the top pole, take him by it a few strides and then ask for it. Like a rollback. Good inside bend, inside leg, and getting that precision is going to make him look a ton better.
Ahh I hope all that makes sense...I kept pausing, typing, pausing, typing...xD
Pssh.I didn't pick up the wrong lead
It's called a counter canter...
...A very advanced maneuver.