Peltier circuit design for TEC
Looks potentially OK. Transistor drive may be too low.

Place a reverse polarity diode across fan so it can freewheel during PWM off.
Fan needs to be DC brush motor and not brushless (BLDCM) for PWM control. .   

You should say what "problems with PWM" mean.
What is desired Peltier current.
See other answers here re driving Peltiers. Olin has things to say about it :-).

Better diagram copy is needed. 
I assume diagram is available at better resolution and has been munted by downsizing. 
Imgur allows files of any sensibly desired size. 

Assume 5A Peltier current.
Assume 5V Arduino.  

TIP122 datasheet here
Beta (current gain) is 1000 min at 3A.
Use 1000 for now.
Vbemax (from datasheet) = 2.5V (!)
At say 5A Peltier current 
Ib = 5A/1000 = 5mA.
Max Ib is about (Vcc - Vbe_darl)/Ib = (5-2.5)/1k = 2.5 mA = too low.
That's wity worst case Beta (at 3A) and worst case Vbe and you can HOPE real device may be better BUT you must always design to worst case.
Making Rb say 220 Ohms or 33 Ohms may help.

Please advise correct figures for assumed value above.