Wireless tool to diagnose why I cant connect to a router...

Hi you guys, i'm a newbie here. Here's my personal problem and challenge...hope you can help me with some tips.

I'm in charge of the wireless network in my work for the past 3 months (basically a school with 3 floors) and right now is working (in theory) with one (main router) 108 wireless ADSL2+ Router TPLink TD-W8920 feeded with a DSL line of 4Mb; to this ADSL router are connected 10 access points: 4 TP links WA5210G and 6 Trendnets TEW 4342APB , to provide free internet access to all students with no restrictions (ordered by my Boss, the principal). Right now, all 10 access points have settings of Open security. Right now the main router is handling the DHCP of all clients, is this healthy?

As you can image, we have almost 20 classrooms in the building (along the 3 floors), so we have one access point every 2 classrooms ( im not an expert, but if you asked me, those are too many access points for one building, right???? can they work together and be efficient?? )

The main problem, in some spots I can't access any of the 10 access points, in some areas I can access just a few or one. So my basic questions are:

a) Is there a wifi tool to know why my laptop is unable to connect to one access point? (a few more details more than "Windows was unable to connect to Aula 101")
2) Any sugestions on improvements would be greatly appreciated!

Saludos from Mexicali!!!!!!