How to manage lot of random yet potential ideas for research and startup?
For your first question, here is the workflow I follow:


Write them down (as @JeffE mentioned)
Organize them in order of viability/ease
Take the most viable idea and do a (very) quick prototype
If prototype is successful, develop it. If not, iterate.


From the prototype you probably have a better idea of how much work it would take to fully develop the research idea. If it seems like you won't be able to handle it in the amount of time you have, or if the outcome is not worth the effort, I usually archive that idea for later. For a  prototype I usually try to do the bare minimum to produce the simplest result possible. A common trap is to start from the complex before you've mastered the simple.  

The second question depends a lot on your personality, so don't take the following advice much too serious. Personally, I have a very hard time focusing on two things at the same time. What I would do is focus on my PhD. You'll surely get many more ideas as your work progresses. For every problem you solve, you'll generate 10 new ones. Be sure to write things down so that you don't forget them later on.

This is a very personal question and you have to see what works for you.