If there were a Coronal mass ejection would having a satellite closer to the sun give us an ability to detect and warn us or would all communications be knocked out and/or the solar flare would be the same speed as the communications?
This answer addresses the overall effects of coronal mass ejection and solar flare events rather than simply the e-m radiation.
The answer is yes, it would help. If budget were not a concern it would be most helpful to have a belt of solar orbiting satellites close to the sun and further out.
Some background terminology:
A CME is a release of plasma - charged particles and also involves variation of magnetic fields. A solar flare is broad band e-m radiation, though people may concentrate on visible or x-ray components. The two phenomena are closely enough linked to cause some sloppiness in usage and so when one is mentioned it can usually be assumed that the discussion could be about the release of particles or e-m radiation or field disturbance.
Communications are not necessarily "knocked out" by the phenomena travelling at light speed. Causes of disturbances to satellites or ground infrastructure could variously be the solar ejected particles or field disturbances that have in turn stirred up particles resident in the Earth's belts.
The answer - more detail. I believe the causes of communications loss on Earth are thus mostly at risk from the slower moving phenomena taking 1-3 days to arrive at Earth.
Satellites have been positioned at the Earth Sun L1 point, such as SOHO, ACE and DSCOVR. These are able to sample interplanetary physical parameters beyond the magnetosphere though this point is actually relatively close to Earth and so its benefit as an Early warning is modest in purely temporal terms (no disrespect intended here to folks working on these programmes - I think they are very interesting and helpful in scientific terms).
Lastly, why would it help? It would enable operators of satellites, polar flying aircraft (and their passengers) and power stations to make better preparations to reduce their vulnerabilities.
As Wikipedia states, an astronomical unit is roughly the distance from the Sun to the Earth. Roughly because the earth's orbit is slightly elliptical.
One astronomical unit is defined as 149,597,870,700 metres.
Any visual information about the Sun travels at the speed of light, which is 299,792,485 m/s. The time taken for light to travel from the Sun to the Earth, even the light from a coronal mass ejection will be:
149,597,870,700 / 299,792,485 = 499 s = 8 minutes & 19 seconds
By placing a probe closer to the Sun, we won't be gaining much time.
Yes, and in fact, it has been done. SOHO, for one, was placed at the L1 point, and studied CMEs, partially to give advanced warning of particularly strong ones. STEREO, which wasn't much closer to the Sun, expanded the ability to see CMEs that were beyond the view of Earth, allowing for better predictions. Also, being able to see them in 3-D assists with this view.