Do you know a tweaker to reach it? Is there any trouble to set it to more than 5 minutes (300 seconds)?

I didn’t experiment with it.

If this value is different from the avast! setting, which will prevail?

The avast setting represents the actual network timeout value. The OE setting doesn’t make much sense when mail goes through avast. OE first sends the mail quickly to avast and then waits until avast sends the mail out. During this whole waiting OE increases its timeout counter. If OE reports timeout it doesn’t necessarily mean that network is not responding, only that the whole mail takes a long time to send.