Chains are used in both push rod and OHC engines (BMW and Saab are examples on the latter). Chains tend to make more noise, especially when they become worn. (basic engineering)
Many high revving motorcycles use chains for the cams, but at high performance applications gears are common. This introduces however more noise.
The service interval on belts have become a lot better lately, 60k is very low. But the cost of a belt is low and many people can do it theirself if they want (not that much more complex than to change waterpump/generator/servopump/AC belt). A chain lasts longer, but not until the end of engine life (unless you have a low quality engine), and it's more expensive to change.
Both chains and belts are stretched somewhat during their life.
Both push rod and OHC engines need oil to the top of the motor, escecially with high performance engines when valve spring cooling is important.
The difference in size of a push rod and DOHC engine is very small, the increase in performance more than compensate for this.
The largest valvetrain problem is probably hydraulic lifters, which can cause wear of too high pressure against the cam or start to leak and the cam to lifter clearance till be off.