We would need to hear a demo of your speaking and your singing voice to try to evaluate you better.
I really enjoyed the vocals on your link of the greatest bass singers. I can't sing anywhere near that low!
The lowest bass singers I have known have very large diameter necks. This makes for a longer set of vocal cords, which, like on a long scale bass guitar neck, creates the possibility of lower fundamental tones, purely by the natural physics and acoustics of the space inside the body of the singer. A long neck also can have similar low frequency acoustical characteristics, because of the length and shape of the vocal tract.
I don't think you should look at your gift of a low voice as being a handicap. You may have been bullied by classmates, but that is usually a case of ignorant and intolerant people ganging up on someone just because they sound different. In your case, and because of the requests you are making also about your speaking voice, I think we need samples of you singing (including areas you are having trouble with) and also samples of you speaking, reading from a book or something that demonstrates what it is that you are uncomfortable about in your speaking voice.
We're here to help you if we can, so don't worry. We won't bully you here.
It's good that you are taking steps to identify what you feel are problems with your voice, and then to seek whatever options there may be to get you, step-by-step, closer to where you want to be.
If you believe that you can do it, and if you are willing to do the work to get there, chances are much greater that you will move in the direction of your goals. Everyone is different, and everyone has a different set of challenges ahead of them.
Believe in yourself. Accept who you are, but choose to do what you can to bring more happiness and joy to yourself.
It's OK for you to want to extend your voice, either upwards AND/OR stretching it lower. You do need to strengthen your voice and increase your overall range if you want to have more options available to you. Again, please submit sample demos so we can hear just what your current capabilities and limitations are, and we can go from there on making recommendations.