I agree that a better RX and an outside antenna might work but I think the most important thing is a better antenna.
That doesn't mean an antenna with more output, necessarily. You need an antenna that rejects local noise better.
I would recommend a small, tuned loop antenna often called a "magnetic loop"
The easiest and cheapest way, especially if you're trying to use a small handheld radio is probably this antenna:http://users.erols.com/k3mt/hla/hla.htm
If you have good results with it, you should build a sturdier version with wire and a capacitor, like this one:http://n3ox.net/projects/rxloop/
A small tuned loop will overcome some of the disadvantages of a cheap RX... it will add some extra frequency selectivity because it has very sharp tuning and it won't have so much output as to overload the radio.
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Your fundamental problem appears to be noise pickup and buying a $1000 HF RX isn't going to fix that. A balanced antenna like a small loop coupled to a small handheld radio running on batteries might.
The most important thing, though, might be figuring out what is causing your noise and shutting it off!
I have that RX loop on my homepage because I built it while living in an apartment so I could try to hear some DX on 80m. It worked very well for nulling some massive noise I had. I found a very sharp null in one particular position, and kept using the loop until one night when it finally clicked... I looked at the position of the loop and the layout of my apartment... followed the null to the next room... and found that the noise culprit was MY OWN DESKTOP COMPUTER.
Don't let this happen to you. Make sure it's not your own TV, computer, or other appliance that's causing your interference!
Make sure you unplug any candidate devices. Sometimes they make noise even if they're turned off.
Now, it's possible that your noise is something harder to take care of, like power line noise or a neighbor's plasma TV. In that case, the small tuned loop could be a great solution. Just put the null on the neighbor.
A new radio might enhance the HF listening experience overall, but if you can't find and eliminate the noise I hear in your file, a new radio won't help at all.