I've been gyming for around three and a half months now. Started when I was 123 lbs and have gained significant weight, 141 lbs to be exact. Height is 5'9. Been eating a lot of calories obviously. I lift heavy and workout five times a week, 1 hour maximum. Q1. Any particular reason for this rapid weight gain or is it normal? Q2. Plus my tummy is getting out too in the process, is cardio recommended during the bulking phase, if yes, how much? Since most of the trainers recommend to wait for the cutting phase to incorporate the any sort of cardio.
You've gained about 18 pounds in about 14 weeks, or about 1.5 pounds per week, which is a reasonable rate at which to gain weight while "lifting heavy", especially since you were (and still are) quite under-weight for your height.
I don't recommend trying to lose bodyfat yet. Consider continuing to build strength and bodyweight until you aren't able to keep adding weight to your exercises during each successive workout without reducing your total volume (that is, sets and reps), or until your bodyweight reaches 190 lb or so, whichever happens first.
It is difficult, but not impossible, to build muscle while losing fat. Typically, the body has either concurrent net increases in both or concurrent net decreases in both. We use strength training to skew the anabolic/catabolic processes toward building/maintaining muscle.
As an aside: If you are truly "lifting heavy", then five workouts per week are too many. Your body needs time to recover and adapt to the stresses that you apply to it in the gym.