Learn Farsi: من این ویدیو را در پروفایل خود ذخیره کرده‌ام.

For compound verbs, only the second word changes. The first word ذخیره stays exactly the same. To make the present perfect of the second word کردن, you take the past stem کرد, add ه to make the past participle کرده, and finally attach the personal ending for 'I', which is ام.

Why is خود used here for 'my' instead of من?

خود means 'self' or 'one's own'. In Farsi, when the person owning the object is the same as the subject of the sentence (in this case, I am saving the video to MY own profile), it is very common and natural to use خود instead of من or a possessive suffix.

What is the purpose of را after این ویدیو?

The word را is used to mark a specific direct object. Because you are talking about a specific video (این ویدیو - this video) that is having an action directly done to it, you must follow it with را.

Do the two parts of the compound verb ذخیره کرده‌ام have to stay together?

Yes, usually. In the present perfect tense, the non-verb part (ذخیره) and the conjugated verb part (کرده‌ام) sit right next to each other at the very end of the sentence.