Is it possible when doing weight training and running?

Hard workouts, with weights or a hard run, will have your muscles hold on to water, for repair and glycogen storage. This can make it hard, especially in the early phases of an exercise program, to actually see what you're losing. I am going through this now, getting back into a regular running program. The day after every long run I am up 2 lbs., even though my calories were well in line the day before--it's fluid retention that comes off slowly over the next few days. You are not yet putting on muscle pounds, though you're on your way. Try to eat well (both caloric targets and quality food) and definitely keep up the workouts, and you will see a weight loss trend, though you'll still have water fluctuations. It's irritating when you want so much to lose weight, but keep in mind that pretty much any day-to-day weight loss (or gain) involves water changes.

Not at all--I am saying the gains and losses you are seeing are mostly water and you can't change that up-and-down unless you stop working out, which isn't a good idea. Just be Zen about the numbers on the scale and eat as much as you're supposed to to fuel your body through the workouts (eat your MFP goal plus your exercise calories, in the best quality food you can manage).