Learn Hardcore Turkish: Ayakkabı ayağımdan kayıyor.

Because you’re talking about one shoe (or a pair viewed as a single unit). If you insisted on plural shoes, you’d say Ayakkabılarım ayağımdan kayıyor (“My shoes are slipping off my foot”).

What case is ayağımdan, and what does it express here?

Ayağımdan is in the ablative case (marked by -dan/-den), which usually means “from.” Here it literally means “from my foot.”

How is ayağımdan built up morphologically?

It consists of three parts:

ayak (foot) → stem changes to ayağ- due to consonant mutation
-ım (1st-person possessive) → “my” → ayağım
-dan (ablative) → “from” → ayağımdan

Why does k in ayak become ğ in ayağ-?

This is a regular consonant-softening (lenition) in Turkish. When you add a suffix that begins with a vowel to –k, it often turns into –ğ. So ayak + -ım → ayağım.

What tense and aspect is kayıyor, and how is it formed?

Kayıyor is the present continuous form of kaymak (“to slip/slide”). Formation:
• kay- (root)
• -ıyor (present continuous suffix, harmonizing to -ıyor)
• zero personal ending for 3rd-person singular (it)
So kay + ıyor → kayıyor (“is slipping”).

Why is there no explicit subject like o ayakkabı or bu ayakkabı?

Turkish allows pro-drop, meaning the subject can be omitted if it’s clear from context. Here kayıyor already implies a 3rd-person subject (“it slips”), and ayakkabı upfront acts as the topic/subject.

The word order is Ayakkabı (subject) ayağımdan (ablative) kayıyor (verb). Is this the normal Turkish order?

Yes. The default order in Turkish is S – O – V (subject – object/adjunct – verb). Ablative phrases function like objects/adverbials and appear before the verb.

Why not say Ayakkabım ayağımdan kayıyor with the possessive -ım on the shoe?

You can! Ayakkabım ayağımdan kayıyor means “My shoe is slipping off my foot.” Leaving out -ım (Ayakkabı ayağımdan kayıyor) simply treats “the shoe” as understood (context makes it clear it’s yours). Adding -ım makes it explicitly “my shoe.”

Could you use a different structure to express the same idea, for example with a reflexive?

Not naturally with kaymak. You might say Ayakkabım ayağımdan çıkıyor (“My shoe is coming off my foot”), but that’s a different verb (çıkmak means “to come off”). For slipping, kaymak + ablative is the idiomatic choice.