This eMac is old, when you opened up the RAM access area and installed the new RAM, did you notice a small half-height AA battery behind or above or below the two RAM slots?
This half-height AA battery is called a PRAM battery. PRAM stand for ParameterRAM.
This battery is responsible for keeping all of the user settings and preferences for different things you setup up on your Mac when your operating system is running (things like setting the date and time, desktop screen savers or pictures, how your system is generally setup, etc.).
If this is the original battery or it only has been replaced once, the PRAM battery only lasts, at best, 5-7 years.
I am thinking that you are having these difficulties because this battery is probably dead or almost dead.
When this battery dies, all sorts of weird system anomalies can happen.
I would like to elimate all hardware issues.
I need you to replace this battery.
Radio Shack has these batteries, but they charge way too much for them (over $20 dollars).
OWC has these batteries for $4.00, but you will have to wait a few days to get this
Here is a link to OWC and the PRAM battery that your eMac uses.
http://eshop.macsales.com/item/Newer%20Technology/BAA36VPRAM/