Most likely what you've described is due to being out of condition and it will get better as you get more acclimated to the exercise. I've had the same problem, even when very well conditioned, and make the transition from swimming to my bike or from the bike to running when training for a triathlon. As you get acclimated it will probably improve or go away completely.
As far as the 2nd question...sure it's ok. Is it optimal? Nope. But will it work? Well for getting in shape and burning
fat - absolutely. For adding slabs of muscle? Yeah, but not as well.
Oh and as far as looking like a hardcore bodybuilder...it's simply not gonna happen. One reason, you username is sweetpgrl8...that means that you are either a female and don't have the hormones for it or a male using that username...again...simply don't have the hormones for it.
If you only have 15 minutes for cardio then I'd suggest you either do it with higher intensity first thing in the morning BEFORE breakfast OR move it to right after your weight workout.
How many days a week do you train? How much time do you have to invest in your workouts?
Here's a sample workout that works well for a few of us on this board and can be adapted to your individual situation.
Squat, deadlift, bench press, row, and overhead press.
Do these on M, W, F. 1 set of each using a weight you can handle easily for 8 reps. When you ca do 12 reps easily with this weight...add more weight...in the smallest increment possible. When you are done with this - go do your cardio.
Want more specifics? Post up more stats and of course what your diet looks like.
Oh yeah, and what you posted will also work fine. Just be sure you are getting a couple of days off each week.