Question: I'm trying to understand the proper way to make up missed salat. For example, say I've missed Zuhr salat, and I reach the masjid as they're praying Asr. Do i make up Zuhr right after Asr, or after Maghrib?
Answer: If you have five or less prayers due, then you must observe their order. In the example you gave, you must compete Thuhr before joining Asr prayer.
Question: What is the proper way to do the sajdah when you know you've forgotten something in your prayer?
Answer: for details, see this article.
Question: Is Witr prayer mandatory? Say I delay it to the later part of night to do after Tahajjud, but I don't wake up. Do i have to make up Witr?
Answer: Yes, you must pray it before praying Fajr if there is still time left to make Fajr on time. Witr is a duty, but not of the same level as the five daily prayers.
Take a look at articles on these topics also.