That is correct - there is one ticket to enter the city and separate tickets are needed to climb both Macchu Picchu and Huayna Picchu mountains. The separate tickets must be purchased combined with the ticket to the city.

No, most people do not, not least because on average it takes an additional 3-4 hours round trip; it is another ~650 meters higher (a moderate hike when dry - difficult when wet), it is not as iconic as Huayna Picchu, and it requires a separate ticket/additional cost (although it is a minimal $5US). Although the tickets for MP mountain are limited to two timed entry groups of 400 people (max) each, in most cases you can buy tickets the day before or even the day of. But to avoid surprises or disappointment I suggest purchasing well ahead of time.

[The same cannot be said for the much more popular Huayna Picchu - it is currently sold out through August and about 50% sold for the next month after that.]

Also note that tickets are sold with individual visitor's names on them to prevent reselling and theft. Visitors must present their passports at the main entrance so their names can be matched to their tickets.

I didn't climb MP when I was there last month so cant offer any insights on the climb itself.