There's certainly cheaper places to house servers. They were probably a big cramped already, and expecting to need that space as they hire more (they aren't limited to their current job listings).
Not to say that's Dropbox's reason, but in my experience many, many large companies host servers in places that are far from being the most cost-effective option.
Bloomberg used to have (maybe they still have it) a data center at 525 Market Street, a skyscraper in the heart of the Financial District. Sometimes cost effectiveness isn't the sole option companies use, maybe it made sense for Bloomberg to have servers close to their customers.
The financial sector isn't the best counter-example. Every little bit of latency matters in HFT, so there's probably a competitive advantage to running servers there compared with the countryside.
I doubt they'd use such expensive property for a data center.
No, you don't put servers in office space. The infrastructure requirements are drastically different.
I mean its not unheard of for a company that size to build out a hardware lab on site in what over office space they're leasing. My previous employer did that through 3 office moves (+10 racks of test gear).