I have skimmed through this thread and offer the following;
I started off with pretty much the same machine you have and I too ran flux core wire for awhile. Also like you, I had a wide variety of issues of poor performance with it; sometimes it would weld decent and then not so much. First thing I did was go to gas and got rid of the flux core, if you are doing car (especially body panel work) you will be way ahead in several areas, first is that you will get better penetration, plus the flux if not properly cleaned off will come back and haunt you when you paint it. Going to gas was a huge improvement, however it still was spotty overall. I was down at my local welding supply place getting something and talking to the guy who worked there, he told me that my gas regulator was probably a POS and that I should get a new one, so I did and it made a HUGE improvement to my welds.
The power thing is real but IMO it is as much the machine as it is the cord, they do not like any drop in power/amperage unlike a Miller or Lincoln which can take the fluctuation a bit better. I ultimately bought a Miller 140 Auto set and then later a 212 Auto set as well as a Diversion 180 TIG (I do a lot of welding). I realize your budget is tight (most of us have been there) but if you are going to do a fair amount of welding, you really need decent equipment. That HF unit is good for someone who welds a few small things a few times a year, but not so much for any real use, they don't have the duty cycle for it.
As to wire, I run .023 in my 110 machine and .030 in my 220 machine. I have found there are very few things I cannot do with the .023 and it is very handy when working on cars, especially body panels. I use the big machine for things that need a lot of penetration.
One thing I can share about technique is tip distance; you need the right distance from the material you are welding or you get very bad results. On smaller wire you need to be closer but there is a limit as to "too close" and "too far" for all of them, you have to find what works for your machine and the wire you are using.
Pushing v Pulling: you can do either type however pulling gives you better penetration whereas pushing gives you a flatter weld (generally)
Wire speed - you need to spend time to determine which speed works best for the type of welding you are doing, the size of wire you are running and the thickness of the material you are welding. Do not believe anything written on the machine or in the book it came with, they are merely guidelines. You must play with the speed for the specific job you are trying to do. It is not necessarily a fine line but there are limits to too fast and too slow.
Differences in practice v actual job - I suspect you are practicing on a bench or something wherein the practice material is laying on something, this will act as a heat sink, thus your welding will be different when you work on the actual job which does not have a heat sink.
Grounds - you have to make sure that you are connected to a solid, clean, good ground and that the cable connected to your machine is connected correctly (on both ends). This was one problem I had early on, the machine would weld good, then bad, etc. and I discovered that the ground cable had loose connections.
Clean material - MIG can weld through quite a bit of crap, however you will find the cleaner the material is the better the weld is. This is ESPECIALLY true if you are welding anything that has ever been galvanized or powder coated. I use flap wheels to clean all my stuff before welding unless it is just a peice O crap and I don't care about it.
Lastly - check all of the connections on your machine, might require taking the cover off but you need to be sure.
I don't know if any of this is helpful but I hope at least something is.
