Without the "te" the sentence would be "Which shoes are you going to put", and that makes no sense so you'd need to have some word to say where you're putting them. The "te" makes it "Which shoes are you going to put ON YOU", which makes a lot more sense. The "te" is linked to the poner, and just makes it be so that you put the shoes on yourself. I hope that made sense.
Could "What shoes are you going to put on?" also be correct. My answer was marked as incorrect. "BBC Learning English" states: "when we are choosing between just two or three options, we usually prefer which. If there is no limit to the number of choices, what is used." In this sentence, we have no idea how many pairs of shoes there are to choose from. "BBC Learning English" goes on to say: "Before nouns what and which can be used interchangeably to ask questions about people or things: What / Which colour trousers would you like? Brown, green, blue, orange or maroon? Which / What writers have made the biggest impression on you?
I think the distinction is more of a nuance here. Which is a more appropriate translation because the selection does have an implicit limitation. So by following your rule stated, although no specific number is provided, it is implied that the shoe choices are controlled in some manner, even if simply by the wardrobe of the subject. Thus, it is more like saying "Which shoes (from the implied, but not exolicitly stated selection) are you going to wear?" Instead of the more general, "What shoes (out if the entire selection of the world) are you going to wear?"
The "yourself" is kind of implied with verbs like ponerse when it's translated.
So, ella se pone la ropa best translates to "she puts her clothes on"
Same with "ducharse" - to shower (oneself) Me ducho - I shower, not necessarily I shower myself
Despertarse - Se despierto a las sies: He wakes up at six (not 'he wakes himself up at six')
There are dozens of verbs like that where the "yourself/himself/herself/themselves" is just left out of the translation.
It could be either "llevar" or "ponerse", depending on the context. "Llevar" is the most common, but "ponerse" is appropriate when you are talking about the moment in which an item of clothing is put on.
So in this example, imagine a couple getting ready for a big night out. The husband has a pair of nice leather shoes, and some tatty old runners.
They wife might say "¿Cuáles zapatos te vas a poner?" or "¿Cuáles zapatos vas a llevar?"
(note that there is no "te" required in the second sentence)