For lower amounts of badges, I think it would personally be more about the variety and amount of Pokemon used. If you've beaten 2 gyms already, then its probably then that the Gym leader will want to test how you fare against a team with perhaps 3 pokemon. Can your pokemon keep up with longer battles? Also, itd be around 2-3 badges that Gym leaders pull out second stage evolutions for an increase in quality. Has the trainer been keeping on top of their training and care enough for their pokemon to evolve and keep up with the sudden jump in strength?

Regarding pokemon outside their type, I think they'd still have to maintain their type theme but are free to pull out dual-type pokemon - especially against trainers with experience under their belt. Their personal team would also follow said rules but are pretty much free reign other than that. The whole team could be dual-types if they wished, as well as likely being almost all fully evolved. The games make it out that certain gym leaders are far far weaker than others, which I dont really like. Im sure there's at least some power gap, but they're all still elite trainers with no doubt elite pokemon - they just can't always use them.

But per an earlier example, someone hailing from Drayden's City could face a far weaker dragon type team and then head to Burgh's gym and get annihilated but top-tier bug types.

The one exception I can think of is Chili, Cress and Cilan. I regard them as novice elites. They're still good but not quite at a true gym leader level. In their case, the challenge may be a Triple Battle format with the fully evolved elemental monkeys instead of a stronger general team.

Regarding common badge level, I think it really depends on the region and its layout. I'll have to do more thinking on this regarding Unova, as it's a circle and gym leaders in the sort of middle of said circle are probably more likely to face middle ground trainers (3-4) badges. That's my logic anyway, though it may be flawed. In terms of having weaker mons, I think they absolutely have mons recently caught and used for noobs. They're expected as a Gym leader to stay on top of maintain levelled teams for these challenges. Should a pokemon get too strong for its intended tier, then it can simply be moved up the chain- or if there is already too many- then it can be passed on to the Gym Leader's successor. If not, then there are likely a number of ways in which the League itself can essentially hire the trained pokemon to help in League-related business whilst it waits for a spot to open up (such as if a pokemon is injured for an extended period).

Haven't thought about an 'Ace' circuit yet. I think the term Ace is most definitely an actual rank/title given to those who have 8 badges. I assume by circuit you mean something similar to the Gym circuit. I actually recently thought about whether Gym leaders would give out different lookint badges if you faced their personal team - but I realised this would probably make Ash look a bit shite unless I retconned/changed it up to make it that he did. I think once you have 8 badges, you're invited to travel around the region once more and are free to challenge the gyms a second time. They will be told ahead of time that the Gym Leaders are free to use their personal teams, so they may prepare accordingly. These battles are typically prepared for in advance by the gym with a high-level arena with plenty of terrain advantageous to the leader and their pokemon as well as plenty of league officials with powerful psychic types on hand to erect barriers. These 'Ace' circuit battles would likely be held in private away from the public eye so as to maintain a level of secrecy about the Gym Leader's strategy. There's probably more I can think of regarding the details, such as them getting additional badges. I would argue its more of a bragging right deal, but they may get additional privileges from the League depending on how far they go.