The NEC only requires two ground rods if one doesn't suffice the 25ohms requirement. If an additional ground is installed then a minium of 6' spacing is required between them. The only time the neutrals and grounds are allowed to tie together are in the main disconnect.
To keep the neutrals and grounds separated, a 4 wire should have been used to feed your sub panel.
Four Wire Feeder
If the panel is supplied by a four wire feeder (hot, hot, neutral, ground), then yes, the ground and neutral should be separate in the second panel.
If the panel is GFCI protected, and fed by a four wire feeder. Mixing the neutrals and grounds in the panel, will almost certainly cause the GFCI to trip. By mixing grounds and neutrals, you've essentially bonded the grounding and neutral bus bars together. This causes neutral current to be split between the grounding and neutral conductors of the feeder. Which means that the GFCI device will not see enough current on the neutral, and will open the circuit.
Three Wire Feeder
If this is an older installation, where the panel is fed by a three wire feeder (hot, hot, neutral), then the neutral bus bar should be bonded to the ground bus bar. In this case, mixing grounds and neutrals is poor aesthetics, but not a code violation.
If the panel is GFCI protected, and fed by a three wire feeder. Mixing grounds and neutrals should not trip the GFCI.
Best Answer
Neutral and ground should only be bonded at the service equipment. This could be at the service drop, the meter, or the service disconnect (250.24(A)(1)).
Looks like the neutral is bonded in your main panel, via the bonding screw.
The second panel is then fed using four wires, to keep ground and neutral separate.