Regarding wearing shoes:
It is permissible to pray barefoot, or to pray wearing shoes.
It is better to pray sometimes barefoot and sometimes wearing shoes, according to what is easy; one should not make it difficult to put shoes on or to take them off in order to pray If a person is barefoot, he should pray barefoot and if he is wearing shoes then he should pray wearing shoes, except when there is a reason not to do so.
If he takes them off, then he should not place them to his right; rather he should place them to his left, if there is no one on his left, otherwise he should put them between his feet. There is a subtle hint that he should not place them in front of himself. This is the etiquette which most worshippers ignore, so you see them praying facing their shoes! This is what was narrated in the saheeh reports from the Prophet (peace and blessings of Allaah be upon him).
The du'a for sujood:
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The prophet said that "The slave is closest to his Lord when he isprostrating, so increase supplication [in it]." You can perform your supplication at any time during the sujood; it doesn't matter when you do it.
Source: https://islamqa.info/en/13340