I have Cooper S 2009 R56.
The car hasn't been used at all for 2.5 years. I finally decided to try to start it. I checked and I need a new battery. But, before I buy a new battery I'd like to start the car and make sure the engine runs and there are no serious issues.
I removed the battery completely. Can I use a jump starter (with battery being removed) to start the car that hasn't been used for 2.5 years or I should start it only with a battery?

Yes you can as there is no difference from permanent hookup. Just leave the loose battery hooked up for a good 15min run time unless there are issues.

Put the battery back in and jump it. I wouldn't jump start a car with no battery in it. Especially one with the sensitive electronics our cars have in them.

Like bagpype stated.. Jumpstart with the battery. Don't let loose cables hang around that can short out expensive parts, such as the alternator or ECU. I'd put cables on the running car (both Neg && Pos on battery), and Pos on the dead battery, and the Neg on the engine block (or side motor mount). It's not really a smart thing to connect the batteries directly. Let the running car charge up the dead car for about 5-10 minutes, holding the RPM's at about 1500-2000 RPMs during that time to get max amperage.
Personally, I like to see if the battery is taking a charge, and trying once with the jumper cables disconnected on at least on post of one of the batteries. Reason for that is, if you can avoid starting with the running car running, it will save you from possibly burning out the alternator in the running car. If that don't work, then let it charge a few more minutes, shut the running car down, then immediately try the non running car to start with the cables fully connected. Again, this method will save from potential damage to the running car's alternator.