When a battery has a bad cell if will typically read about 10.5 volts, a bad cell produces no voltage or little voltage. A good, fully charged battery should have around 12.6 to 12.8 volts at the terminals. Now, you replaced the starter, so if the battery is good (have it checked at the parts store) then I would check for power at the fuse block (ign) and also check the voltage drop there. However, you wires in a push button directly to the solenoid which rules out ignition. So, test the battery, if it's good, clean the terminals real good with baking soda and water, clean the cables as well, make sure they are tight, check your grounds, then make sure you hooked up the new starter correctly (and makes sure it's connections there are clean) then see what it does. Wiring in a remote start switch (the button) eliminates everything but battery, cables, and starter. So it's one of those 3, honestly it sounds like a bad battery, but may be cables. Btw, I have seen several bad batteries that wouldn't jump either, I've used jump boxes directly to cables disconnected from battery to jump cars before...it's not the best way, but in some cases it's the only way, after all a jump box is a battery.