Learn Finnish: Äidiltä on turha pyytää joustavaa kotiintuloaikaa. - It is useless to ask for a flexible curfew from mom.

In Finnish, when you ask for something from someone using the verb pyytää, the person you are asking is put in the ablative case (-lta or -ltä). The structure literally translates to asking from someone.

Could I use the verb kysyä instead of pyytää to say to ask?

No, because they have distinct meanings. You use kysyä when you are asking a question to get information. You use pyytää when you are making a request to get an object, a favor, or permission. Since this sentence is about requesting a flexible curfew, pyytää is required.

Why is the object joustavaa kotiintuloaikaa in the partitive case?

The verb pyytää takes a partitive object when the thing you are requesting is abstract, uncountable, or indefinite. Since a flexible curfew refers to an abstract concept of time (aikaa), the entire noun phrase must be in the partitive.

The current lesson is about sikäli kuin (insofar as). How could that conjunction fit into a sentence like this?

You can use sikäli kuin to add a limiting condition to a definitive statement like this. For example, you could add it to the end: Äidiltä on turha pyytää joustavaa kotiintuloaikaa, sikäli kuin minä hänet tunnen. (It is useless to ask for a flexible curfew from mom, insofar as I know her.)