Intermittent No Start: Need Help!!!!!

If anyone can help greatly appreciated. I have a 2003 rsx base model 5sp.
I've recently run into trouble with my car randomly not starting but a couple of pumps of the clutch and turning the ignition and it used to fire up back then it would randomly happen maybe once in a month. Now that the problem is less intermittent and a regular thing it makes driving around almost impossible.

To give a bit of background, when it wouldnt start if you put jumpers on it, the car still wouldnt start (no crank no start). I would have to wait and hope it would come back on. We did a test and it came up the battery had a bad cell so we replaced it and it was fine for a whole day. The next day same issue, i had a friend come over and try and search out the problem with me, he determined it to be the black/white wire going to the starter solenoid it was just hanging inside the engine bay. As he tried to connect it the head of the wire broke, so we soldered a new wire to it and connected it. The issue was still there. However if one person dis-connects the head of the cable and reconnects it as someone else is turning the ignition the car will fire right up, and will continue to do so for about a day. Having someone play with the wire while someone turns the ignition or bump starting are the only ways to get the car to start after the no crank/no start. After this it will fire up for about a day until it does it again. Though yesterday when i tried to reproduce the symptoms it took me 2 full warm ups and 2 cool downs to get the car to not start. But as soon as i went to put a test light on it, it fired right back up. I'm stumped, i've taken it to a few shops and they said they cannot help me until the car doesnt turn on at all. Can anyone provide ideas?