Now that you've pointed it out, may i point out the rear suspension of Honda CG-125 (Classic) ? They are Rigid for a reason. And they are REALLY hard as compared to other bikes in its class. (A good reason to call CG125 a VERY UNCOMFORTABLE bike)
CG-125 rely nearly 70% on its front shocks (which is why they are kept too soft), and a mere 30% reliance is on rear shocks. (Their Stiffness helps the bike perform well in cornering and also to keep the bike's balance good on road.
As far as Tyres are conerned, Recently i studied a bit (actually quite more than that) about Tyres, their Designs (aka Guddi), and their Hardness/Softness and different Road/OffRoad/Racing Tyres. I was actually going to purchase a new tyre for my CG-125 but was confused A Lot between different tyres available in the market. And i've come up to a conclusion that BOTH Front & Rear Tyres SHOULD be of same Design for best Road Grip & Performance in nearly all terrains.
For reference, check out the Servise Long Life 3.00-17 6PR (CG-125's Rear Tyre), or Panther Power 6PR tyre ... notice their designs. They are Teh Best tyres you can use for Crusing Around in city or for Long Routes, and they CAN work well in nearly all weather conditions ... Too Much Dirt is an exception. (Well i guess you shouldn't make a Classic CG125 a MotorCross Bike :p)