Experience the game however you see fit.

Relying on a single unit hogging all the exp is a viable tactic. IIRC the speedrun of the game relies heavily on Adell.

In Disgaea games, unique characters (especially DLCs) are usually on par with tier 4~5 generics in terms of stats, and have unique skills. You may use only generics to make the game more challenging. Also, this will make you more prepared for Axel Mode, where you can't use any unique character from Adell Mode, including DLCs.

You can lose to Axel to start a new cycle (NG+), similarly to losing to Mid-Boss in D1.

Summoning Experiments are how you unlock the DLCs. The battles may randomly have large exp rewards in the bonus list, which may be why you feel overleveled for the point you're at. The DLCs were originally all lvl 100, with matching bonus list rewards, and were changed to have a level corresponding to the chapter you're at, but the bonus list rewards hasn't been adjusted in the process. Maybe don't use DLCs? Maybe progress the story some more and see if it's challenging enough for you?

And yes, stronger enemies bills may help.

There's no real call to start over since the first few chapters are easy regardless of what characters you use, but it sounds like you'd get more out of the game by starting to build your party out of mostly creatable characters. The ones from summoning experiments didn't exist at all (as playable) in the original Disgaea 2 and were added as DLC in the PSP version, they're about as strong or slightly stronger than Adell and Rozalin, so using only those is basically like having an entire party of main characters. Most notably they have much better stat growths and weapon mastery compared to tier 1 generics, so very unbalanced for the early parts of the game.

As for soloing maps, in Disgaea 1 the best strategy was to have a single super unit, and while in Disgaea 2 a lot of maps are designed to encourage the use of multiple characters, it doesn't change the fact that stats is king and if a character has enough stats, nothing will really be a threat. Which is to say soloing is still powerful, all exp from kills go to one character, you only need gear for one character, and it just escalates. You can still play effectively (as in: without extra grinding) with multiple characters though, thanks to stuff like the combo multiplier and getting more stuff from the bonus gauge.

Anyway, if you're looking for a good challenge... this game has the Dark World maps, which are good to start about when you beat the main story (lv 70-90-ish characters), and going through the stages in order it progresses the level/difficulty scaling until you have about level 300 characters with rank 38/39 gear, at which point enemies can go as high as level 1000. The maps usually have a lot of prohibitive rules or quirky mechanics, which are (usually) much easier to get around with a varied team of evenly leveled characters than with a single beefier one.

Finally, about the stronger enemies bill... it makes enemies a bit tougher in the short term, but since it works by incrasing their level you also get bigger rewards from killing them (exp/money/mana), so in the end you'll be getting ahead faster. And just raising the level doesn't help the enemies that much. Cuts down on grinding though so I guess that's good.

(the stats I mentioned for main story/dark world assume no stronger enemies bills)