I'm testing this with the following version of Opera:
Version 6.01 Build 1041 Platform Win32 System Windows 98
Animated gifs made with JASC Animation Shop version 1.03
Both bugs became obvious when I looked at my existing "New" animation, on my home page http://imaginatorium.org/. Bug 2 could be ignored, but bug 1 makes a very ugly mess, which does seem to be Opera's fault.
This is a reproducible bug: it can be avoided by avoiding completely transparent rows at the edge of an animation, but this makes it impossible to have an animation which "grows". The details unfortunately seem to vary rather arbitrarily, so the detailed descriptions in (( double parenthesis )) keep changing, and probably aren't true.
The following animated gif (50 x 25 x 64colours) has a complete border which is transparent. A ring of droppings (spurious pixels) appears around the edge:
This version has a 256-colour palette: 
A single frame with the same transparent border gives no problem: 
Going back to the problem, simply setting a pixel in the top row to nontransparent on a single frame ((clears the top row of problems, but produces more droppings on the right and bottom sides)): 
Setting a pixel in the rightmost row to nontransparent on a single frame ((clears the right and bottom, but leaves a bizarre orange pixel in the upper right corner)): 
Opera appears to treat any iteration count greater than 1 as meaning loop indefinitely. (Various droppings on the edge of these.)
Loop count 1: 
Loop count 2: 
Loop count "infinity": 
This page put together by Brian Chandler April 2002