YOu are describing a bad battery and/or 'dirty' power to the vehicle electrical system.

If your battery is a white "Varta" battery - replace it. DO NOT TEST IT, replace it-today-now.
If your battery is over three years old and the following list does not help, replace it.

On to the list.....
There is a little box near the battery where a pigtail from the positive battery terminal goes. This is a distribution point for several items in the car - pry the cover open and verity the nuts are all tight.

Verify the battery terminals are tight and clean - remove them from the battery and brush the posts and terminals with a steel wire brush if they look bad.
Follow the battery ground cable to the bolt that attaches it to the fender - is the lug clean? Is the bolt tight? If either battery cable's terminals are really bad looking, it could be that the corrosion has set into the crimp and you need either new cables or at LEAST new cable terminals.

Open the box next to the battery that contains the ECM/PTCM, RCM and BCM - is there ANY evidence of water in there?

Look under the passenger side floor mat for water intrusion.

Look in the trunk/hatch area for water intrusion - paying close attention to the PASSENGER side.

If you find water intrusion in any of these areas, you are about to spend a LOT of money. Sorry, but that's the way it is.