Learn English – Any problem with ‘Who did open the door?’ Justify please

They're both grammatically correct, but they don't mean exactly the same thing.

The normal way of expressing have to in the past is simply had to, as in your second example sentence. For example,

Today I have to go to the office. Yesterday I had to go to the store.

Your first sentence, "I did have to play football", looks like an example of the emphatic do. This is the use of to do as an auxiliary verb to give emphasis to something. For example, if someone said to you, "...but you didn't have to go to the store yesterday," you might respond, "No, I did have to go to the store yesterday." Or you might say, "I didn't have to play football in school, but I did have to play tennis," to emphasize the contrast.

Note that the negation of "have to" in the present or past always uses do, as in

I have to eat

vs

I don't have to eat.

or

I had to eat

vs

I didn't have to eat.

but in a positive sense, like "I did have to eat", do is not normally necessary.

The writer maybe sees push open as a phrasal verb, like put on. Phrasal verbs can be used in two ways:

1) She put the dress on
2) She put on the dress.

Your sentence matches form 2). I cannot find any dictionary evidence to confirm that it is normal to use push open as a phrasal verb, but this NGram indicates that quite a few writers use this form.

Here are a couple of examples:

When you've finished putting beads on all the pins, open the big (#3) safety pin and use the nail file to push open the little loop-the-loop at the end of it My best friends and me

Silence promises danger not peace, for always the sounds that push open my door or your door, are remembered as coming out of nothing. In the shadow of sharpeville