Blue Coil. At the bottom of the logic board, near the speaker. Check the following picture, and identify whether that blue coil is present on your board or not. http://i01.i.aliimg.com/img/pb/960/865/7... It is commonly damaged during repair, and causes loss of signal./

Sounds like the little knobbly bit on your home button flex has deteriorated (not uncommon) first step is to change the home button flex cable (ebay it) and make sure you follow the guides on here. In the mean time you can use a feature on the iPod called Assistive Touch to press the home button. http://appleinsider.com/articles/11/07/1...

A picture of the soldered connection would give us a better insight to if it's bad or good. Have you considered the new battery may be faulty?

Have you checked the LCD connector? It can sometimes pop back out if it wasn't snapped in properly. If you have already stuck the screen down then heat it up and take your time prying it off again to check.

Have you tried restoring it? Failing that has it been exposed to any liquid?

Have you screwed the metal mid plate down with ALL the screws and the correct screws in the correct holes? Also check you have removed all glass shards and there is nothing sitting around the edges of the iPod. Use something to scrape lightly around the side to be sure. Check for dents in the corner of the iPod. These are the biggest cause of screens not fitting properly. If a corner is dented or the frame is bent then you need to bend it out. Also make sure the long black cable for the digitizer connection is not folding under the LCD by hanging over the metal edge, this will not allow it to sit. It must be folded in an S shape and sit nicely next to the LCD connector.

Sounds like the lead-free solder has cracked securing the battery connector onto the motherboard. These will need to be re-soldered professionally or if you're handy with micro soldering you can do it yourself.

Take the logic board out and clean it thoroughly with isopropyl alcohol and a cotton bud. You will need to remove the metal shields covering the processor and what not - corrosion hides here and causes all sorts of mischief.

Water damage? Simplest thing to try is a new battery. They are cheap and childs play to replace. Guides are on here.

None working would indicate a damaged IC chip or water damage. Either way you will need a professional to repair it unless you want to have a go yourself. There are two mics in the iPhone so first step is to find the guide on here and replace both - if that fails your IC has gone and will need replacing - something for a professional.