It's complicated. Normally for us no it wouldn't make sense. You would plan to amortize the hardware cost over something like 3 years, but you need to have the $ to buy the hardware and also know what hardware you'll need for future growth. The unpredictability is why cloud usually wins, but cloud is expensive in exchange for flexibility.
In our case here I'm donating the hardware and it's ridiculously overpowered for what we need, so hosting it somewhere makes sense.
Basically. It wouldn't be with them directly, but with a smaller provider I know that has several racks. They'll rent me a space in one of their racks, based on our power and network requirements.
Most colocation providers don't want to deal with sharing multiple customers in a rack, they also don't generally provide network access.
Imapsync works well, have uses it hundreds of times on massive accounts. Not with Gmail specifically though. I'd imagine you would lose labels.