I am interested in picking up a musical instrument which can also help increase my lung capacity and vocal projection. Trumpet, trombone, and saxophone are some of the instruments cross my mind, are there any other/better instruments that fit this specific purpose?
Are you a singer who is primarily looking to increase your vocal performance or are you actually interested in learning the instrument for the instrument itself?
I would say these two things are different. Because if you are really only interested in singing, then there are plenty of vocal exercises you can do to increase all of the things you want and much more without needing to learn a special instrument. Also, if you are mainly interested in singing and you do also want to learn an instrument, it might be more beneficial to learn and instrument that you can play while singing (like guitar, or piano, etc).
As far as vocal exercises, if you are willing to pay, you could find a private vocal coach locally who could work with you on diagnosing and addressing your specific problem areas. On the free side, Youtube has many great vocal coaches with all kinds of vocal exercises that you can work on by yourself. Here are some great resources I've seen on youtube for vocals:
Eric Arceneaux: https://www.youtube.com/user/EricArceneaux/videos
Kevin Richards (for rock vocals): https://www.youtube.com/user/RocktheStageNYC/videos
Sophie Shear: https://www.youtube.com/user/SophieShear/videos
Those are some great places to start. I've seen several tutorials from all of them and they are really good.
I can't help with choosing between trumpet/trombone/sax/etc, but I thought I'd offer some alternatives if your main purpose for this question was to get better at singing.
There are rather few synergies between the mechanisms of playing most instruments and singing, to the degree that it makes no sense to pick up a particular instrument except for the sake of playing the instrument on its own.
Lung capacity does not really change all that much and it is rarely a limiting factor in singing: it's much more important to focus on efficient use of air. In terms of breath support, the oboe is likely one of the most taxing instruments. However, the kind of pressure and air management are quite different from that of singing: the best you can hope for is to become more aware of the breath support mechanisms and give the respective muscles a bit of workout.
It's like swimming in order to train for running: same muscles, but hardly useful for training one with the other once you are bothering about more than general fitness.
These are some instruments that can help you increase your lung capacity:
- stationary bike
… if you play them 4–6 sessions per week at medium to high intensity.
They will also help with projection by improving your endurance, relaxation, posture, and diaphragmatic power and control.
If playing a wind, reed or brass instrument could improve your singing voice, then this would have been a standard part of vocal training for centuries. But it never has been, and it is not. It seems that you have imagined this idea yourself.
Learning an instrument while you learn to sing is certainly a good idea from the standpoint of becoming a well-rounded musician. But there is no such thing as learning to play and instrument having the secondary benefit of giving you a stronger singing voice.
If you want a stronger singing voice, take voice lessons from a voice teacher and practice a lot.
And while you are at it, if you smoke tobacco, or anything else, quit now. If you really want a strong and powerful voice and lung capacity, smoking is the long-term enemy of that goal.