Monthly schedules at Emirates vary quite a bit, and can vary a lot depending on whether you are on the Boeing or Airbus (I'm a Boeing guy).
All pilots are assigned to one of five bid groups. The groups rotate in their bidding order every month, meaning that you will be 'top bid' once every five months, and therefore 'bottom' bid once every five months as well. What sort of trips you fly obviously depends much on what month it is as well as your personal preference on flying. Emirates uses the Ad Opt preferential bidding system, so some of your monthly roster depends on your knowledge of the mathematics of the system and bidding savvy. You can plan on being on a month straight short call reserve for two months of every year.
A typical roster will have you flying more than you probably would in the US/Europe/Australia. As for me, I fly around 75-85 hrs per month with anything from 8 to 16 days off per month. EK is only obligated to give you 8 months off during the month and sometimes that's exactly what you'll get, although they list some days as 'rest' days so the actual monthly total of days spent in base is somewhat better than that. Generally speaking, trips to Europe go 'senior' and trips to the far East/Australia and turns go 'junior'. The longest trips in the company are on the Boeing. They are nine days long and go through Australia/New Zealand. Not sure what the longest trip on the Airbus is- perhaps 5 days? European trips are usually 24 hrs off at destination but some of the Asian and Australian trips give 48 hrs off downroute. Lots of the flying in Emirates is in the middle of the night, particularly for turns, which is why nobody seems to want them much.
I think the Airbus guys tend to work a bit harder than the Boeing guys, particularly those that are dual qualified on the 330/340. Airbus guys tend to do more turns and work more days per month, although their trip lengths aren't as long as on the Boeing. More and more Airbus flying seems to be shifting to the Boeing as more 777's show up so I suspect as time rolls on the Boeing and Airbus rosters will tend to equalize.
As far as leave goes, you are given 42 days off per year contractually. With some agressive bidding on your top bid month you might also be able to pull off another two or three weeks per year off. I don't think it'd be unrealistic to be able to get home 3-4 times per year as long as you don't spend weeks on end at home on each visit.
Hope this all helps.