Questions about our 2008 Prius (104K miles +)

Hello, Prius owners! I am new to this web-site and am still getting used to our Prius, so, please, be patient with me We have purchased our 2008 Prius in October 2012 form a Chevy dealership (I know . The sales man pretty much had NO idea how to turn it on, how to drive it, etc. So we had to figure it all out on our own...For example, we thought that we had to put our key in the slot every time (because we were told so) until we realized that our key battery was dead, which brings me to...
Question #1 - After we went to a Toyota dealership, they changed our key battery and were told that from then on we didn't have to put the key in a slot and that the car would unlock if we got close and lock if we left with the keyes...Well, that hasn't been the case so far. Sometimes it does unlock when we get close, but not all the time. And it certainly doesn't lock when we leave (I believe I would've heard the locking sound). So, what is the problem here?
Question # 2 - Then, our Prius started to act funky and would not turn on sometimes...My husband went to do an oil change to the same Toyota dealership in November and had our Prius checked out. Our small battery was tested at 50% battery life and to replace it we would have to pay about $400 (grrrrreat!). We didn't want to rush with purchasing a new battery since a. we just bought the car, b. this payment was not budgeted and we could not afford it...and still can't. So, a few times we had a very hard time starting it (especially, in Canada this winter). Recently (few weeks ago) we couldn't start it AT ALL and had to jump start the battery. After that, my husband told me that he would have a hard time starting it once in a while (he would start the car - the red triangular sign comes on, but the "ready" sign doesn't, and he would turn it off and on again). Pretty much most of the time when we turn it on, the red rectangular sign comes on and disappears. Obviously, all of this has caused a great concern to us and some disappointment. So, is all of this happening because of the small battery that needs to be replaced or can there be other issues?
Questions #3 - Away from the heavy questions...Since we didn't get the owner's manual with the car, I have been wondering - do we HAVE TO turn the lights off before we turn the car off or can we leave it on since they turn off automatically with the car? I'm kind of concerned about killing the small battery with that. So, yes, we do make it a point to turn it off.
Question #4 - In the begging when we just bought it, whenever we would drive the car, the large battery would very rarely show that it was charged until "green" level, almost always it would be blue. Recently it started to show "green" much more often...I also recently drove with someone who has a 2005 Prius and his battery was "green" almost the whole time we were in the car. Why would this be happening? is it related to the smaller battery not working at 100%?
Question #5 - When we bought the car, we got the regular report about the car where it showed all the check marks that the car was in good condition. So, how come the battery was at 50 % life condition when we checked it a month after the purchase? Is it because they never properly checked it?
Question #6 - I just recently noticed that the back/trunk window has a spot (a few red lines) where it doesn't defrost the window. So, it works everywhere besides kind of in the middle. Is that the way it's supposed to be?
I apologize for the lengthy questions. I do appreciate any help or input. Thank you!

 

thanks for the answers! wow, i really didn't know much about our car i'll printing the user's manual and reading it!

 

thank you for your response! i downloaded the manual and will be reading it tonight!

 

thank you! how can i listen to the starter? in my case, the car does turn on, but sometimes from a second try. today i had a "p" warning came up saying that i need to park on the flat surface (i always park in the same parking lot). i got this warning before and had to turn off, then turn on the car...

 

thanks! i'm sorry but i don't even know what "exide" or "optima" mean is that the battery options? thank God, i don't have any fluctuations in charge (very high to very low), just mainly it's in high "blues".

 

thank you for your input! we just had the battery checked at 50% in november...so, i'm guessing it just got worse from there

 

thanks! very informative! so, are you saying that we can manage to install the new battery ourselves? we really need to purchase it asap! can i ask a regular mechanic friend to do it?

 

thanks! yes, i tried touching the inside of the handle, and from a second or third try the car unlocked the driver's door. i think i need to try it again. i did know about the outside button and even our child can lock it

 

great, thanks! i hope we can handle it we've never changed a battery before

 

sounds easy! we'll definitely try! i have zero experience with this, especially with an electric car