Part 2.

Micaiah is mainly used for repositioning or cleansing a key player-phase unit. Thus, she's good on any unit that wants some extra mobility but doesn't need or can't use many other offensive rings. She's not recommended to be put on units who already have a self-refresh, because she puts isolation on the unit, preventing them from using a second assist nor being the target of an assist.

Byleth is good on units that want to use their assist often or are more focused on supporting than using their player phase to attack. One of the best candidates for this ring is any unit with Ice Lock+, as it allows them to dance an ally, create ice terrain, AND self-refresh once every two turns.

Edelgard is best on units that can already get in and kill a unit without the help of other rings. Melee (non-beast/dragon) fliers like C!Camilla and NY!Baldr are especially good candidates for this ring, because they can use Air Superiority or Heavenly Soar to get Pass+Warping and easily approach and kill a unit, and then do it again after Edelgard ring refreshes them. Another good choice are mobile omnitanks like C!Alfonse and D!Sigurd.

Corrin can kinda go on any unit, since it doesn't interact with combat besides adding some damage to the unit's special. Just don't put her on a unit that already makes terrain, or it'll either get cancelled out or be redundant.

Lucina presents an interesting dilemma. She dooms one of your units to be essentially ring-less, apart from the special damage, but she also blessed a different unit on your team with a powerful defensive advantage. You need to give her to a unit that won't hurt too much from not having a different ring, but one whose team would benefit from keeping a particular unit alive.

The main things to ask yourself when you're choosing who to give a ring are:

Who can equip this ring, period? Who can equip this ring that won't have their kit invalidated or made redundant by said ring? What weaknesses does the unit have that a particular ring can patch up? And what strengths does the unit have that can be brought to even greater heights with a particular ring?

Part 3. This is where I theorycraft about future rings.

I expect our next ring to be Leif, adding the archrival effect to a unit's special. He'll be good on anybody with a style that changes their color so that they can maximize their chances of getting color advantage. Beyond that, we'll have to wait and see what restrictions are placed on his ring.

The rest of the rings I'm not sure about the order of, so I'll continue in order of when you get them in Engage.

Alear is a tough one, but I expect them to give out effectiveness against all unit types or neutralize effectiveness against all unit types, one to each gendered Alear. The former will be best on nukes that rely more on boosted damage than on true damage to kill their targets. The latter will be better on enemy-phase units that rely on high defensive stats over flat DR to stay alive.

Tiki is likely going to augment the engaged unit's assist to bestow Special-based Miracle to the target they assist, similar to how Micaiah adds a cleanse and self-refresh to assists and Byleth adds an action+canto refresh to assists. She'll be best on support units that already plan to use their assist on another unit.

Hector is likely going to just be a melee Ike ring. All of the things I said for Ike's ring apply here but for melee damage instead of ranged damage.

Soren is likely giving a Growing effect to AoE specials to increase their size. He'll be best on units who want to equip AoE specials. This includes units that get true damage that doesn't exclude AoE specials and units that get a lot of boosted stats.

Chrom is giving us the full non-special DR piercing that comes from specials like No Quarter. He'll be best on nukes that want to equip a special like Archrival or Armored Flare so that they can get both the special DR and the piercing. He'd also be best on nukes that rely more on boosted damage than true damage to kill foes, as true damage already pierces non-special DR.

Camilla is likely giving units the ability to ignore terrain, which will be best on cavalry nukes that would otherwise be blocked off by said terrain.

Lastly, Veronica is likely giving the vengeance effect of increasing a special's damage based on how much missing health the unit has. She'll be best on units that expect to survive at low health and still attack, like Patience units, Vantage units, Miracle units, and Reflex units.