These are a major part of the user experience and will provide access to some of the most important features in each section of the app.  I'm sure this is still a work in progress, but I found the current system a bit mind-boggling, since the buttons (symbols) do not consistently represent their functionality from context to context.
As *sets* of buttons go, I'm not sure this toolbar is 'one size fits all' given the places it shows up.  Here's what I take to be the current state of things, with problematic buttons (for instance, those whose function changes from place to place) highlighted in red.  Some suggestions follow.
Online Game View:
+ button = Next (game)
Action button = brings up Action menu
Flip button = flips the board
Magnifier button = Chat
? button = analysis board
Arrow buttons = forward/backward thru move list
Play Computer:
+ button = Next (game) - i.e. new game screen
Action button = brings up Action menu
Flip button = flips the board & switches sides
Magnifier button = analysis board
? button = Hint 
Arrow buttons = forward/backward thru move list
Tactics Trainer:
+ button = Next (problem)  Sometimes the new problem I get is frozen, but I think that's the intent here.
Action button = brings up Action menu
Flip button = flips the board
Magnifier button = analysis board 
? button = Show solution 
Arrow buttons = forward/backward thru move list
Analysis Board:
+ button = does nothing
Action button = does nothing
Flip button = flips the board
Magnifier button = display FEN 
? button = does nothing
Arrow buttons = forward/backward thru move list
 
Here are my suggestions:
Online Game View:
I think this is the important one.  Move the analysis board function to the magnifier button so it's consistent with computer games.  (The magnifying glass is a good symbol for the analysis board - not so much for chatting.)
Move the 'chat' feature to the position of the current "?" button, but change the symbol to something that represents chatting, like a little speech bubble.  A speech bubble that doesn't look like a magnifying glass.  :)
Play Computer & Tactics Trainer:
This suggestion is less important, because a "?" is arguably a good symbol for getting guidance in both contexts.  However, maybe consider using a light-bulb for the computer-play hint and stick with the question mark for revealing the solution in Tactics Trainer.
Analysis Board:
Just use the Flip button and the two Arrow buttons here.  Keep it simple. 
A pop-up of the FEN is of no use at all.  (If if copied itself to my 3.0 copy/paste clipboard, we might be on to something, but even then I'm not sure how valuable it would be...)
Consider maybe adding a new, third button to the Analysis Board tool-bar:  a << button that restores to the 'current' position for folks to lazy to hit the single Back arrow multiple times.
 
Cheers!