Cardio + Dumbbells

You're probably not eating enough, especially if you're not eating back your exercise calories. This could mess with your metabolism I think. Also if you've been working out at the same intensity level and with the same weights for the past two months, your body is probably used to the exercise and not burning as many calories.

From what I been reading Goldmay might be right... everything I read says a person shouldn't go below 1200 calories

http://www.shape.com/healthy-eating/diet-tips/10-things-you-dont-know-about-calories

Measure yourself as in measuring your neck, waist, and hips. You might find yourself building muscle (which weighs more than fat) you might notice you are getting smaller that way than on the scale.

Eating under 1200 calories, not even the slightest chance ever that she has built any muscle at all. Muscle also doesn't weigh more than fat, it is just denser (takes up less space.)

You are exercising 6 days a week cardio and 7 days a week weight lifting. Your TDEE (didn't know age so put 35) is around 2500-2700, you should in no way be eating less than a net of 1500 calories a day with that amount or exercise or you will have literally no fuel. Even 1500 calories at that level of exercise would be a very aggressive attempt at losing weight, I'd aim for a pound of fat a week which would be around 2000 calories a day at your level of exercise.

I am burning almost 500 calories biking every day and then the weights for about 20 to 30 minutes.

You are therefore netting around 500 calories a day, do you really think that is enough for your body to run properly on? Yes, I am a male, younger and bigger but I am currently eating 2500 calories to lose 2lb a week. You need to start netting a lot higher than 500 calories or you are very unlikely to see any progress.

Measure yourself as in measuring your neck, waist, and hips. You might find yourself building muscle (which weighs more than fat) you might notice you are getting smaller that way than on the scale.

Eating under 1200 calories, not even the slightest chance ever that she has built any muscle at all. Muscle also doesn't weigh more than fat, it is just denser (takes up less space.)

You are exercising 6 days a week cardio and 7 days a week weight lifting. Your TDEE (didn't know age so put 35) is around 2500-2700, you should in no way be eating less than a net of 1500 calories a day with that amount or exercise or you will have literally no fuel. Even 1500 calories at that level of exercise would be a very aggressive attempt at losing weight, I'd aim for a pound of fat a week which would be around 2000 calories a day at your level of exercise.

I am 32 btw. I should've told my nutritionist how much exercise I was planning to do, I barely mentioned some indoor biking to her. She just told me to be around 1200 to 1300 calories each day.
I can feel my muscles stressed and tired, you are probably right with the fuel thing.
Readjusting my diet can get me burning more fat?
Should I eat back all the calories that I burn by exercising?

If you use myfitnesspals calorie thing eat whatever they suggest and also ALL the exercise calories. The number they set you to eat is already creating the deficit, there is no need to go under your calories that it suggests. I used to eat 2000 calories to lose weight and then I stalled for ages, upped my calories and started losing again. I also feel a lot better when I'm eating more calories, I have more energy and I can always hit my macros. You will feel a lot better which means better performance in your workouts and you have enough fuel to keep you going

Try this link:

http://scoobysworkshop.com/calorie-calculator/

Plugging in your info yields these results:

BMR: 1650 - most on here will tell you to eat this as your MINIMUM

TDEE at moderate exercise (3-5 hours/wk): 2557

Recommended calories: 2046 (TDEE minus 20%)

Whoever told you to eat 1200 calories wasn't doing you any favors, IMO. Try keeping your exercise regimen and eating around 2000 calories. Since TDEE already takes into account your exercise, you won't "eat back your exercise calories" in this regimen. MFP counts on you eating back your exercise calories.

Steady state cardio causes metabolic adaptation especially if you do it every day. Switch to HIIT

1200 cals is too few
skip the weight lifting

The weight gain is from glycogen/water retention from starting exercise. Normal.

Bioelectric Impedance scales are inaccurate at reading precise BF% because water in the body can change the readings.

And you're also probably eating too little overall.

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Been in fitness for 30 years and have studied kinesiology and nutrition