OK, let's answer some of this, sort of out of order...
Not only will a cheap mixer cause noise, there are definite sonic differences between something cheap and something that costs more. And the difference there comes from component quality. Cheap stuff (like Ammoon, Alto, Harbinger, et al) cuts corners on components, with the result being looser tolerances, which sort of cascades as your signal path goes through the board. One sub-par component is bad enough...now consider what a couple dozen of them in an audio chain will cumulatively do. Plus, certain mixers have a very specific sound quality to them, most notably the English-designed/made ones. This is what makes a pre-Behringer Midas desk so desirable...but not so much a post-Behringer one, as these don't have the same rounded "English tone" anymore.
Computer line-ins aren't the right thing to use, nope. The culprit here is noise at your A-D conversion stage. This is due to the A-D on a typical sound card (or sound card on the motherboard, depending) being typically unshielded from in-case electronic noise, plus the fact that that connection is going to be a consumer-level (-10 dB instead of +4) line-in and it's also unbalanced, which tends to allow more electronic crud into your signal chain. The line-out isn't as problematic, but to get really good results on recording, you need an outboard interface that's +4 dB, takes either XLR or 1/4" TRS balanced lines, and has a proper ground. And one other point: everything in a recording setup should be star-grounded. By this, I mean that everything you use needs to have a ground that is the same as all other devices, usually done by grounding everything to a single ground point (hence the name). By doing this, you can lower noise and help avoid ground loop issues.
QuickTime is not only the wrong tool, it's also VERY out of date. Use a proper DAW. You already have Audacity, so try recording in that instead. I actually multitrack in Ableton 10.0.6...but I chop loops and clips and also do my final editing and normalizing in Audacity. It works better for that, while Ableton works beautifully on multitracking, track comping, and so on. Ableton is also not the only choice; you might look at Bitwig, which is similar but has some of its aspects more streamlined than Ableton Live.
Now for the last pile of questions...first, EQ. Technically, there's three types: parametric, graphic, and program. Parametric is the type where you can specify the frequency per band, the level at that frequency, etc; you often see these on mixing desks in some form or another. Graphic EQs are the ones with fixed frequency bands with level controls, and tend to see more use in live applications for room correction, but can also be useful for similar purposes in the studio. And program EQs are things such as Pultecs, where you have specific boost/cut stages with their own tailored frequencies, often also working on the overtone spectrum of the selected frequency. This last bit is very typical of the Pultec EQP-1A's low end cut/boost control, where the 'boost' also works on the overtones of the selected frequency, but the 'cut' acts like a normal shelf, with the -3 dB point at the frequency.
For the most part, a program EQ is the only EQ you should boost levels on. All other equalizers should be used to subtract from what's present in the raw signal unless you're using the EQ as an effect in some way. The reason for this is that it's easier to compensate for lower levels of something in a mix than it is to correct levels of some type that're too hot. For example, let's say that one track has a band in the lower-mids that's sticking out, a sonic 'lump' as it were. It would be easier to isolate the 'lump's' frequency and reduce that on that one track than to bring everything else up in various levels and bands to even out the 'lump'. But with a program EQ, what's being done is more akin to "sculpting your mix's tone color"; accordingly, most of the time you'll see program EQs on the final mixbus to do those timbral adjustments.
However, tinkering with EQ without a good monitoring chain...flat, unforgiving response from as low as is feasible in the bass all the way up to the ultrasonic...is basically pointless. It's like trying to read a map, but you've forgotten to put on the reading glasses you need...ergo, you're probably going to get lost. Never skimp on monitors...unless, of course, you're trying to check your mix on a more "real-world" equivalent, in which case you need to incorporate those "everyday" monitors alongside the other, more precise ones. And this, btw, is how you check your mix; if you need to know how something sounds on, say, a typical set of computer speakers, by all means use some of those after you've done your mix on the mains. But if something needs fixing as a result, do that work back on the mains again. Motown studios always had a pair of 6"x9" car speakers in some cobbled-together wooden boxes in their studios specifically because Berry Gordy wanted to know how their stuff sounded in your typical car...and of course, Motown stuff sounds great in the car because of this "check". Headphones, however, are not something you mix in unless you're specifically mixing for headphones.
As for compression, there are again several types. Limiters basically "smash" everything above their threshold level and hold the dynamic limit right there. More typical compressors have various (and often adjustable) settings for how aggressively the compression happens as the desired level is approached, plus what sort of degree of compression (ie: ratio) is needed. And program compressors, like their EQ counterparts, are more for riding gain and "gluing together" a mix while used on the mixbus. As for the right way to use these, first keep in mind that anything over 4:1 ratio winds up behaving and sounding like limiting, especially with a hard "knee" (that "aggression" setting) at the threshold level. To get a compressor to behave transparently, use lower compression ratios and softer "knee" settings, which will then allow the compressor to compress over-level signals enough to fix level problems but not to make the track in question sound like its being "mashed". Unless, of course, that's what you want, since compressors are also useful for adding distortion and overload character to sounds that can use a beef-up.
Program compressors, though...those are a bit different. In their case, you use compression to get the overall stereo level on the mixbus to "float" around the desired track's loudness without exceeding 0 dB. So the meters on a program compressor might be floating at around -3 to -5 dB, and you'll use the makeup gain to bring that result's level up to where you need it to be post-compression. These are a good bit trickier to use well; like anything else in music worth doing, they require practice.
As for what to use...that's up to you, and what sort of sound you're going for. A good place to start, though, would be KVR Audio (https://www.kvraudio.com/), which has a trove of free plugins. You should, over time, be able to find the ones that work for your music and workflow...but again, this takes time, because in this process you're actually tailoring your DAW to be your bespoke recording "instrument".
Hopefully some of that is of use...