If you are not ready, then you are not ready, however april/march is plenty of time to get in shape.
You do not have to run in the morning, you can run at night or even during lunch time or at the weekends, whilst running is the best accessory work for boxing, it is by no means essential, cycling, swimming, skipping all have similar benefits.
As you get fitter, you will naturally lose weight, do not sweat the small stuff, if you are working hard in the gym and avoiding junk food and soda, you should lose weight.
Boxing is a hobby for most people that participate, train as much as you want as long as you are having fun.