do i need a new battery?

Ok -- so I think I have isolated the problem:

So I purchased a new battery ... and it remote started successfully in the cold once or twice .. but reverted to the problem as previously described:  It will crank a couple turns on the first morning cold start .. then stop cranking .. the parking lights go on, and it "thinks" the car is on... b/c the next time you hold down the remote start button, it chirps twice and the lights go out (as if it just shut itself off).

Now .. i'm thinking this could be narrowed down to two problems:

1) bad ground (less likely -- it was working perfectly until I did the following about a week ago -- right around the time my problem started)

2) Right around the time this started.. I installed the defroster feature onto this Maxima.  Now, ultrastart does not have a dedicated defroster trigger, so here's how i did it -  I employed two relays and the constant to momentary diagram from this website:

First relay: 85 = ignition wire on remote starter (blue on the Ultra 1270TL), 86 to GWR wire on r/s unit, 87 = ignition wire on r/s unit (blue, again), 30 = connected to 85 on SECOND relay. 

SECOND relay:  85 (as previously described), 86 = chassis ground, 87 = connected to defroster switch (i.e. the wire off the defroster switch that needs to go to Ground to trip the defroster) and 30 = 1K capacitor in parallel with a 10K ohm resistor (as per the diagram on this website for constant to momentary) and the negative end of the capacitor/resistor combo goes to chassis ground.

Basically... I only wanted the defroster to go on when remote started, and not each time the owner started the car manually (i.e. key in ignition).  And I wanted to create a "momentary" ground to the defroster trip wire so that I wouldn't burn out the defroster, utilizing the constant 12volt+ from the ignition wire to trip a series of relays to create a "momentary" ground output to the defroster switch.  When it's not cold, it works flawlessly.  However, now that it's cold, it's perhaps .. robbing the ignition wire of too much power? .. and could this be causing the brain to think the car as started?  Or, once the capacitor's charge is full -- the blip in current messes with the "brain" enough to cause it to just do something haywire?

Any input would be appreciated.  Thanks for all the replies -- very helpful info.

Ok --

Update:  So I disconnected my defroster Relay#1 from the defroster unit, and on the first cold start of the day .. same problem:

User presses " * " button, car chirps once, car cranks for 1.0 seconds, car stops cranking, parking lights come on and stay constant. 

User then has to hold down * button to shut "off" remote starter, and then when she presses * again, it chirps once, cranks .. only THIS time it starts no problem.

Why doesn't it try three attempts at starting?

PS:  I've never "learned" the tach in the frigid, first morning cold .. I've always learned the tach when the car was warmed up .. or at least when the engine oil was not jelly.

Should I re-learn the tach in the extreme cold so the remote starter knows to rev the car to a higher speed?

Thanks again (...frustrated beyond belief)

either try a different tach source, or relearn tach, but there 2 different ways to do it, some systems respond better to 1 technique better than the other and vice versa. A: make sure vehicle is warmed up TOTALLY, meaning idling at the lowest possible RPM, turn on blower etc to increase load etc and then learn tach, or B: rev engine slightly as you learn tach, technically not the proper method but sometimes helps.