Russian has 2 types of verbs: perfective and imperfective. Imperfective verbs present an action as an interval of time that has a start and and end. Perfective verbs present an action as a moment in time.
The verbs «пока́зывать» and «показа́ть» look really similar and both mean roughly the same, but «пока́зывать» means 'to be showing (for a certain period of time)', while «показа́ть» means something like 'to finish showing, to successfully show' (and it doesn't have a period of time, finishing showing happens at one moment).
It gets more interesting with the tenses. Basically, only imperfective verbs have the present tense. «Я пока́зываю» means 'I show, I'm showing' (it's a form of the verb «пока́зывать»).
It makes no sense to put «пока́зать» in present tense, because it describes a moment in time. So, when you use the form that's usually present, with the verb «показа́ть» gets a future meaning: «Я покажу́» 'I will show.
So, it needs no auxillary verb: only imperfect verbs need auxillary forms to get a future meaning.
Hope that helps.
"I don't understand how the auxiliary verb will is used here" - WILL используется здесь только для того, чтобы показать Будущее время и ВСЁ! (WILL is used here in order to show FUTURE TIME and it's all, no more functions in this case) Но на русский will show можно переводить и как 1. Покажу и как 2. Буду показывать. В данном случае - Покажу. Но: "Если ты всё время забываешь, где лежит рис, я буду показывать тебе снова и снова, пока ты не запомнишь." :) (If you always forget where the rice is, I will show you again and again until you memorize (remember?) )
It's just my opinion (not an English speaker):
"WHERE is the rice?" is a question while the sentence above is not a question, thus "I will show you WHERE the rice is."
I wonder WHY you didn't say anything. Not "I wonder WHY didn't you say anything."
I just realised HOW grateful I am. Not "I just realised HOW grateful am I."
I don't know WHO I am. Not "I don't know WHO am I."
I'm not sure WHEN he is coming back. Not "I'm not sure WHEN is he coming back."
Basically, if there is another sentence preceding the question, the sentence is not a question, thus the order.
P/S: Correct me if I'm wrong (any native speaker)!