The click/no start (at least in auto-trans models) is usually the ignition switch from what I've read. I recently had a slow cranking issue and a no click/no start that I thought was a worn-out starter, but the replacement did the same thing until I replaced the ignition switch as well. Unplug the signal wire for the starter (grey plug, attached to a bracket on the side of the starter or dangling nearby if the last person who had the starter out didn't have time to mess around with the stupid bracket) and can hook up a test lead to the side coming from the starter. If the engine cranks poorly on the ignition switch, but cranks properly when you touch the test wire to battery +, the solenoid isn't getting the power it needs from the switch. The switch itself (not the lock cylinder) is cheap and surprisingly easy to swap out. (You can also hotwire it right from the ignition switch by bridging the black/yellow and black/white wires, which may be easier than getting to the starter wiring, especially if the bracket's still there.)
The hard starting once it is turning over does sound like an idle control issue, but I doubt the idle screw moved on its own. More likely an idle component isn't working for some reason. I solved a stalling-at-lights-once-warmed-up problem on mine by cleaning the plug for the IACV (tower-looking deal at the driver's rear of the motor) and re-bending the female pins a little because they were loose. You might also have a look at the plug for the other idle valve (the IACV-AAC), which is the one with the screw. I'm not sure why there are two idle valves, but there are.
Also make sure your battery terminals are clean and tight. Doesn't sound like it's the starting issue but it's easy to check, and worth a look if the issue started when you swapped the battery.