Coordinate system dependence of simple charged particle

Two questions, 2nd to be written separately concerns more difficult problem for motion of extended charge distributions. First:

Two point charges q1,q2. Determine motion of q1 using Newton’s ODE. Force is q1*q2/r^2. I assume this gives velocity of q2 relative to q1? I assume this result has coordinate system attached to particle q1? Isn’t this a non-inertial coordinate system? I am assuming motion near earth surface and that this is essentially inertial coordinate system.

If I then attach coordinate system to earth, shouldn’t the speed of q1 be only half of what you get by attaching to q1? What would the ODE be in this case?

Physics 101 text I have seems to always give problems for point charges considering only two charges, attaching coordinate system to one of the charges.