First post, so forgive me if this is in the wrong place.

About a week ago when i tried to start my car after school it made a slow sluggish sound but wouldn't start, after several tries it did and I got it home. Since then turning the key in the on position has lit all lights but putting it in start does nothing but turn on the oil pump (or the fuel pump whichever is supposed to turn on). There is no other sound and no other attempt to start.

Things I've done:
Replace the battery, we killed it trying to fix this.
Attempted to start in neutral, as i understand it that should rule out the neutral safe switch
Follow the ignition switch wire, unbroken until it disappears into said neutral switch
Attempted to jump the solenoid with the battery wire and a screwdriver, unsuccessful.

Nothing appears corroded, At this point im assuming its the starter motor itself, and i just need it replaced, so my questions.

Does my testing prove its the starter motor, solenoid, or both?
At this point, can i rule out ignition switch?
Should i replace both the motor and solenoid if only one it them needs it?

Thanks for any help.