Is a combined BS/MS in math a good idea for pursuit of PhD?
I finished my undergraduate degree at 20, and I know a number of others who have finished undergraduate considerably younger than I did. So, no, 20/21 is definitely not too young to apply for and begin a graduate degree program.

Since you list yourself as a US college student, here are a few questions to ask yourself before embarking on a dual-degree program:


Who is going to pay for the MS part of the program? You may not want to have to pay out of pocket the extra costs of the master's, when it would normally be covered if you picked it up "on the way" in the PhD program.
Are you happy with your choices for a research advisor for your master's thesis? (No point in being miserable by working for the wrong advisor—even if it is a semester or two, a miserable time is not what you want.
Where do graduates of your school's master's program end up for PhD programs? Are these schools you'd want to go to? (You can ask the same question of the bachelor's program, of course.)
Who will write your letters of recommendation? Do you have people who know your work beyond the classroom well enough to recommend you for external graduate programs? If not, this may vote in favor of staying for the master's program. 
Do you have to make a decision about which program to do before you apply for external programs? No point in making a tough choice if you can keep it as a "fallback" option.


As for topics and coursework, I will leave those answers to the mathematicians.