Learn Russian: Давай не вмешиваться в её жизнь!

The pronoun свой (and its forms like свою) always reflects back to the subject of the sentence. The implied subject of Давай is 'we' (you and I). Since we are talking about interfering in someone else's life, not our own, we cannot use свою. We must use её (her).

Why is вмешиваться in the infinitive rather than a conjugated form?

When you use Давай to say 'Let's', you pair it with the infinitive if the verb is imperfective. Negative suggestions like 'Let's not...' almost always use an imperfective infinitive in Russian. So, Давай не вмешиваться acts as a set structure meaning 'Let's not interfere'.

What case is жизнь in here, and why?

It is in the Accusative case. The verb вмешиваться (to interfere or meddle) requires the preposition в followed by the Accusative case. Even though it is an abstract concept, you can think of it as a figurative motion or direction: you are putting yourself 'into' someone else's affairs.

What is the function of the -ся ending on вмешиваться?

It makes the verb reflexive. The non-reflexive verb вмешивать means 'to mix something in' (like an ingredient into a bowl). Adding -ся literally means to 'mix oneself in', which translates perfectly to 'interfere' or 'meddle'.