How can I understand if there's a problem with HDD Connector?

Open it up again and make sure that your battery is connected properly and that you have a good battery. That is for the not turning on without being connected. You could take a look at your HDD cable with a magnifier to see if you have a torn cable, relative inexpensive easy fix. Also make sure you have the cable connected right and that the clip that hold the cable to the connector is okay. You can also try to restore iPod from Disk Mode as shown here http://support.apple.com/kb/TS1463 It does sound and look like you do have some HDD error. I have a few of these models that had a "stuck" hard drive and needed a "slap on the back" and here is a copy of one of my previous answers "Some times when you drop your ipod the lens inside skips. This can also happen with other incidents that are not as violent as dropping it. Reset your ipod by holding the middle and menu button until you hear clicking, then with one hand hold the ipod firmly and hit the side of it onto the PALM of your other hand. Sometimes this jumps the needle back into place but not all the time. It may also require a few attempts since most of us are not willing to treat it like a 1970's TV :) NO GUARANTEES ARE MADE HERE AND I WILL NOT ASSUME RESPONSIBILITY FOR ANY DAMAGE :) There are also multiple threats out on it even answer here on ifixit Why does my iPod show a red "x" in a circle?... and http://www.pinkbike.com/forum/listcommen... I hope this will help you out to get your iPod going again. Good Luck