No, the wire going to the coil would not stop the starter from working.
You would know if you removed or disconnected a grounding wire or strap, when installing the heater hose, I hope. I doubt you did.
I hate to say this, but you really need some local experienced help to solve this. There has to be a friend who can at least help you check to see if the starter will "manually" turn over bolted back up in your car with all the wires hooked back up.
If I were there, I would do the screwdrive trick which would take only a couple of seconds. (I don't recommend you do the screwdriver thing) IF the starter turned over, then I would start looking at the ignition, safety switch, etc. that John Z was talking about.
IF the starter did not turn over, then I would suspect the starter, battery or grounding path from the motor to the chassis. To systematiclly eliminate my suspected parts/circuits, I would have the battery tested, have the starter tested. If thye were good, I would then take jumper cables long enough to run from both battery terminals on the battery, behind the seat, to the starter. Make sure the tranny is in neutral and parking brake is on. Then, clamp the negative cable to a clean starter mount bolt on the starter and touch the positive cable to the small S terminal of the starter. The starter should then manually turn over.
If not, then the battery is indeed bad or the starter needs a new solenoid or rebuilding. Basically, I have replicated what the guy at the auto store did, just with your battery as a power source. It worked at his place but not at yours with your battery.
IF it does turn over, then disconnect just the negative battery cable from the starter mount bolt and touch the positive cable to the S terminal again. If it turns over, then back to checking out the ignition, safety etc. If it does not, then you have to inspect your grounding path for the starter back to the negative battery terminal. This path goes from the starter bolted to the motor, then the motor with a cable or strap to the chassis. Your negative battery cable is also bolted to the chassis in the back, which completes the ground circuit for the starter.
You already know the positive cable to the starter is good, because other items work, like the lights and wipers. Neither of these items depend on grounding from the motor so they will work, even if your starter doesn't.
This is going to probably be something real simple that I managed to make look hard to impossible. Good luck. I have had starter problems in the past and have always been able to fix them. Good luck.