1 No you are not wrong, unless your partner is smashing from 3/4 court, in which case you can peel off and they can push up, but you have to specifically train this so you and partner know when to rotate and how to communicate. Covering cross from rear court smash is not the idea, you should always move across with your partner. If they are right, you are positioned slightly right, if they are left you are positioned slightly left. If you are having to cover cross your partner is playing bad shots.
Or if they are coming out then maybe you are playing bad shots. Look at video 2 1.37 you are leaving a huge gap with that cross court smash.
Video 1 3.19 you should push up after 3/4 backhand smash while your partner peels off. Your partner is right here, you are wrong. Gap has been left front left by you. You also left yourself in bad position for fast return to you. Racket up move forwards, don’t stand and watch.
Also if you are going cross court hard like that don’t be surprised if your partner is in the wrong position because that’s not a great shot selection. Look at the rear right gap. You get caught out here a lot at higher level, go softer or straight.
Video 1 2.27 your partner is out far left so move out left more you are not looking to cover cross here but take off the straight return down tramline so no point being middle. Forget about the cross, no-one is going hard cross from tramline unless seriously high level. If anyone gets it fair enough because it is a low percentage shot. If your partner is not always going for or near the tramline on a smash and you are having to cover them they are playing a stupid shot.
put your racket up at the net!
Video 2, 1.06 push up, don’t stand there.
2 This is context dependent. If you or your partner is out of position go soft. If you are losing flat exchanges go soft. Go soft to mix it up. Go soft if opponent is stood back and slow. But always push up whether you go soft or hard.