Yes, but what is using that CPU%
The resident protection is only scanning files that are active, e.g. they are being opened and run, etc. that is what is meant by on-access, when the file is accessed it will be scanned.
So it isn't any different they use the same signatures and scanning engine, just that the on-demand scan is scanning all files depending on the strength and areas you select to be scanned. Over time as you have routed out these inactive files you shouldn't bump into to this very often.
Sorry I can't help with P2P questions I don't use them at all.
Gmail uses SSL (secure encrypted) email designed to keep prying eyes out including AVs. avast can't currently scan SSL email without something like STunnel to act as a middleman in the process, version 5 of avast should have that functionality incorporated so you can scan SSL email. Gmail does scan its email so the risk is minimal and if you exercise care, don't open attachments or click links in unsolicited email (without checking them out) then that risk is even lower.
Being stuck on safebrowsing.clients.google.com/safebrowsing/download....etc gives me the impression that you tried modifying the Internet Mail redirect ports and added the SSL port gmail numbers, as I said avast currently can't handle SSL to doing that would doom it to failure.
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I think we have gotten to the crux of your original problem, your installation is stuffed (sorry about getting technical), there are 7 providers and normally only 6 running (unless you have MS Outlook not express installed).
So I would suggest a clean reinstall:
Download the latest version of avast
http://www.avast.com/eng/download-avast-home.html and save it to your HDD, somewhere you can find it again. Use that when you reinstall.
Download the avast! Uninstall Utility,
find it here and save it to your HDD.
1. Now uninstall (using add remove programs, if you can't do that start from the next step), reboot.
2. run the avast! Uninstall Utility, reboot.
If step 1 failed it may be necessary to run this from safe mode, once complete reboot into normal mode.
3. install the latest version, reboot.