If you cannot send it back for some reason, you could try pulling out the input jack. It's likely either behind the pickguard, which you will have to carefully take off, or in it's own little socket, which will be held with a couple of screws. Turn your amp off and insert your jack while you can see it. Is everything making contact where it should? Sometimes you can bend in the hot connection (middle part that touches the tip of the cord jack) with your finger so it makes better contact. Given that you have already tried another cord, this is the likely problem. At worst, buy a new jack for $5 and put it in.
So, your tuner... I think it's probably not going to pick up much sound from your electric. That no-cable ability is more for acoustic guitars. You can probably do it, but you'd have to hold the tuner really close to the strings. I would not consider this a deficiency with the tuner, myself. My favorite tuner I have right now is a chromatic clip on type. Clips on the headstock of any guitar and reads the vibration in the neck, not the sound (though it's capable of that).
For your head set, what did you get? You need to get a pretty decent pair of earphones. I had a set that was great for listening to music on the PC, but when I tried to use with an amp, they sounded like total crap. I bought a new pair with the lowest frequency range I could find. It was a Sony set, 5mhz, I think. Even came with a 1/4 inch adapter. I see them on sale almost all the time for aboot $25 CDN. ;-)
I hope your future musical endeavours go better for you. You might decide on the type of guitar you want and read some reviews. Here are a couple of sites I use to check reviews.
http://www.ultimate-guitar.com/reviews/electric_guitars/
http://reviews.harmony-central.com/
I've recently grown partial to the Fender Squires if that is any help for you. :) I got one to save on the wear and tear of daily practice on my Gibson L6 and I dig it way more than I expected.
...so ever since then, I always hang on to the buckle.