Good to see that you got the program. There's no such thing as perfect form, perfect form refers to get the most of the exercise and get as little chance of injury as possible. So yes, it is better to do them with elbows going back. And you sort of answered your own question, your chest is in fact probably stronger than your triceps, because if you do push ups with elbows flaring outside, it uses your chest more, if you do them back, it doesn't use the chest as much so it uses the triceps more. At least that's what I believe is happening here. Well just keep doing them (or if you can't do even one, do negatives or do them on the knees), you will eventually get better at them, dips help a lot as well for push strength.
For the dips, your torso is supposed to lean a bit forward, well there's more variations, doing them straight uses the triceps more, if you lean a little it hits the chest as well, which is what we do here.