Does the same team bonus item have a greater chance to be a rarer item than usual?

Solution 1:

Short answer: No.

Now that this system has been live for a few months, it seems safe to say that based on the lack of evidence otherwise, the drop rate for the bonus items awarded when spinning a same-team gym photo disc is not more weighted towards rarer items than the other items awarded by spinning a photo disc at a non-same-team-controlled gym. That said, while the individual likelihood for any one awarded item to be a rarer one is the same, you do receive extra items, which in turn makes it easier to get one of those rarer items.

It may be worth noting that different drop rates for spinning a gym photo disc versus a Pokéstop seems to be possible, based on data such from threads like this one, which reports roughly a 5% better chance to get a potion, revive, or berry compared to an earlier aggregation of drop rates for Pokéstops, but based on the findings there as well as the comments alongside it, rarer items (like the coveted evolution items) don't seem to be more likely.