What kind of DC power supply has least noise
If you really got a cheap switching power supply, then most of the noise you're seeing is probably not due to a lack of clean regulation, but because the AC lines couple back into your supply lines after the switch mode supply; your observations under "1." match that. It's really hard to avoid that – only a very dedicated device design and excellent measurement shielding can elimate that.

However, there's a couple of things that are often done to overcome this:


Make sure you don't have any power lines running into your measurement chambers. That especially applies to ceiling lights etc.
Make your ground better. If your ground level starts oscillating when you connect your supply ground to it, you simply have a high impedance w.r.t. to "earth".


Generally, if you want to build a somewhat efficient, cheap, low-noise power supply, you're often in for a Switching-Mode Power supply, like the cheap one you're using, followed by a relatively large capacitor that guarantees that load changes and slight fluctuations don't matter that much, followed by a low-ESR capacitor (typically, ceramic), followed by a linear regulator, followed, again, by a buffer capacitor. That way, you can get the "non-energy-burning, fast-to-adapt-to-load-changes" behaviour of the SMPS, and the low-noise properties of a good linear regulator (it might be worth looking a minimal bit further than the usual, ancient, 780X and LM317 regulators – there are a few more modern linear regulators on the market, and some of them have better noise immunity).