Learn Finnish: Minä pyydän sinua näyttelemään roolisi hyvin.

This is the core structure for verbs like pyytää (to ask), käskeä (to order), and neuvoa (to advise). These verbs take the target person in the partitive case (sinua) and the action they are being asked to do in the 3rd infinitive illative, which is the -maan/-mään form.

What case is roolisi (your role) in, and why doesn't it have an object marker?

It is the total object of the verb näyttelemään, so it would normally take the genitive-like accusative ending (roolin). However, there is a special rule: when you attach a possessive suffix like -si (your) to a word ending in -n, the -n is dropped. Therefore, roolin + -si becomes simply roolisi.

The verb näytellä looks a lot like näyttää (to show). Are they related?

Yes, näytellä is a derived form of näyttää. The -ella/-ellä verb ending often implies a continuous, repeated, or sometimes more casual action. While näyttää means to show or to look like, näytellä specifically means acting, performing, or playing a part.

Could I rephrase this sentence using an että (that) clause?