Can I connect ICs directly together
The resistors shouldn't be necessary. Literally hundreds of millions of devices work perfectly with ICs directly connected. I blame the power supply.

Pentium100 makes an important point: especially digital ICs should be properly decoupled. If you don't, switching may cause negative spikes on the power supply, which may cause false triggers in flip-flops. Use 100nF capacitors as close as possible between Vdd and ground.

Then this fishy ground. You shouldn't have two grounds! Ground of the STK500 and your circuit should be connected, why aren't they?