Learn Finnish: Minä muistan sen yhä.

The verb muistaa (to remember) usually takes a total object when you remember a specific, complete thing or event. Because you are remembering "it" completely, "it" takes the genitive-like accusative form sen. If you only remembered a vague part of it, you would use the partitive form sitä.

Could I use vielä instead of yhä to mean "still"?

Yes, absolutely! Both yhä and vielä mean "still" and are often interchangeable. The word yhä puts a slight emphasis on a continuous, unbroken state ("still to this day"), which fits nicely with holding onto an old memory, but Minä muistan sen vielä is also completely correct.

Do I have to include minä, or can I just say Muistan sen yhä?

You can safely drop minä and just say Muistan sen yhä. In Finnish, the verb ending (-n in muistan) already tells the listener that "I" am doing the action. Including minä here just adds a little bit of emphasis, like saying "I still remember it."

Can I change the word order and say Minä muistan yhä sen?

You can, but it changes the emphasis. In Finnish, the most important or surprising piece of information often goes at the end of the sentence. Minä muistan sen yhä highlights that you still remember it (focusing on yhä). Putting sen at the end would shift the focus to the specific thing you are remembering.