What is the difference in gene sequencing tests vs gene deletion/duplication tests? Should one type of genetic test be done before the other?

Depends. Sequencing is like reading war and peace letter by letter, and deletion/duplication testing is like seeing if a chapter is completely missing or present as an extra copy. Which first? Depends on the condition. For duchenne muscular dystrophy, look for deletions first. For brca, sequence first. Your clinical geneticist can discuss the strategy with you.
No, Wrong Issue. The issues are exactly as you have described. When a section of dna is sequenced, sometimes duplicate copies of genes are found within the dna sequence; sometimes genes usually present within the particular dna are not present, i.e. Have been deleted at some point when the dna was copied prior to whatever previous cell division.