Seems like alot of electrical gremlins here are caused by corrosion on the battery terminals. Pull them apart, clean them and put them back togather. Check the grounds for corrosion also. Then check the connections at the alternator, make sure that they are tight and clean. If all is well, check the alternator. If you cannot test it yourself, Auto Zone and many other parts stores will check them for free. Probably a good time to have the battery load tested as well.
Hope this will fix your problems.