The quality of captured wild pets is random. As of patch 5.1, upon entering a battle the Pet Battle UI shows the quality of a wild pet (idicated by a color-coded border around its icon). If you're farming a specific quality, if you forfeit the match your team will incur a small damage penalty (roughly 10%). There is no accurate way to determine a pet's quality prior to the start of a battle.

There are more passive methods to gain wow classic gold, but these are slower. For example, many people choose to fish in places like Tanaris, or kill Harpies in Feralas and loot items such as mageweaves. These are rather AFK-able, but again, not as rewarding. Some people choose to favour fun over anything, and like in any MMO (massively multiplayer online game), the best way to have fun is playing games with friends or your guild. Dungeon runs with your friends provide social interaction and are definitely helpful over the long-term for your gold stack!
They seem to have expected some things, though, see adding pets with aquatic damage (gun and ooze) aswell as a resistant cockroach to stop us from cheesing everything with elementals. Yet, a fact their damage output is low combined with shields not checking for racial damage means this mechanic prevents a lot as you mentioned. The last boss has got an ability to remove basic shields so perhaps that was not in oblivion either… unlike the reactive damage. Also, an ability that forces swap altogether with an ability which prevents from being swapped, including critters? I can't say I like it.
This is something that people really need to start to understand, not just for Classic or WoW but software developement in general. Fixing a bug isn't just flipping a switch or changing a single digit. You have a machine with literally hundreds of thousands of parts, designed by dozens of different people at different times over the course of 20 and more years. You change a single thing and it might break 50 other things in very subtle ways that might not even emerge until months later (i.e. the very high M+ keys some players got at the start of BfA).
Keep in mind that these quests are optional, and are not necessary to continue leveling and capturing companions. They are also account-wide, so if you complete the quest chain on one character, you will not be able to complete it again on another. However, any progress that you've made will be saved and you can pick up where you left off in the quest chain on another character. The dailies that you've unlocked will also be accessible by all your toons.

Only really to expand on Gráinne's comment as regards the 'collecting' and 'roster development' sections - which are, as said 'distinct' although they are intrinsically linked - I would like to point you in the direction of the in-game Achievements > Battle Pets > Collect tab. There are a series of achievements called Raiding With Leashes, pretty much all of which can be completed via the auction house - if you have the gold/inclination - or by completing old raids.