first we need to know if you use or used any other security applications before.
to your second point:
usually avast has no messages which appear when you work on your computer (seems to be an other program)…
what do the messages say? how do they look like?
i was running trend when i installed avast and the pop ups run up from botom of page with a blue border anouncing that avast is scanning the current page i am loading
Go to the thread I highlighted for you above and follow the instructions to turn off the popup in the Webshield.
Your connection problem is almost certainly caused by trying to run two antivirus products together. They will conflict.
You need to decide which one you want and make sure it is the only one installed on your system.