Notebook Boot and Lockup

Some details :

Laptop is 5 yrs Old. / Battery holds charge only for 10-15 mins. (I use my laptop with power adpater plugged in all the time) 

Problem:

When I forget to plug in power adapter, my laptop shuts down after 10-15 mins(obvious). After that when I plug in the adapter and press POWER button screen displays"Enter Administrator Password or Power On Password". I have never set up any of such passwords. Intrestingly, now when I charge the laptop for about 15 mins. and then turn it On/OFF 8-10 times, suddenly it boots up properly and displays Windows login page and works normally. This problem does not occur when I charge battery (to max of whatever capacity left in it)  and then SHUT DOWN. Problem only occurs when battery is drained to ZERO but it somehow starts working normally after turning ON/OFF several times. How and why is this happening is beyond me!

MY THOUGHTS:

After googling I found out that some people are replacing CMOS battery to get rid of the 'Administrator/Power On Passwords'.

As my laptop is "OLD" , Could there be any issues with my CMOS battery? Should I try replacing it?

What could be the problem ? and How can I resolve it?

*IMPORTANT:  I have never set up any passwords in BIOS. and my laptop works when I charge my battery and try to turn On/OFF several times. Point is, I'm not asking to crack/bypass passwords in the forum. Posted same topic few days before and got a reply saying 'HP community forum does not help in bypassing passwords. That is the reason for this lonnnng post. added some context this time'

Thanks in advance!