Breeding compatibility considers the following:
- Egg Groups - Pokemon can only breed if their first Egg Groups are the same
- Gender - Pokemon can only breed with the opposite gender, or Ditto (Genderless Pokémon can only breed with Ditto)
- OT - To have higher breeding chances, the Pokemon need to have different Original Trainers
- Family/Siblings - To be compatible, the Pokemon cannot be siblings
- Species - To have higher breeding chances, the Pokemon must be the same species (not Pokedex number; for example, Lucario and Mega Lucario are not a perfect breeding pair)

Retro, Baby and Legendary Pokémon cannot breed (with the exception of Mewtwo, Deoxys, Arceus, Genesect, Diancie, Type: Null, Silvally, and Calyrex). Manaphy can breed with Ditto to produce Phione eggs. Shadow Pokémon can breed, but will only produce the regular form. When breeding Alolan/Galarian Pokémon, you will not get the regular form, only the regional form. When breeding Unown, you'll get a random one, but the Unown that you're breeding has a higher chance to appear.

The higher the rarity, the lower the breeding rate. (Easy, Medium, Hard, Rare, Starter/Special, Legendary, [in no particular order, since I haven't tested these:] Alolan/Galarian/Unown)

Boosts:
- Premium - +20%
- White/Black Flute - +100%

- Fountain - unknown; very small boost, possibly 5% for 'slightly increased', and 10% for 'increased' (these are just guesses, and not actual rates)

- Valentine's Day (14/2) - unknown; extremely large boost, likely between 100-400%

Breeding messages:
"The two seem to match perfectly." - best possible breeding rate (commonly referred to as a 'perfect pair') - different OT, same species 

"The two seem to get along very well." - same species, same OT / event + counterpart, different OT

"The two seem to get along." - different species, different OT / event + counterpart, same OT / Ditto (OT doesn't change the message for Ditto; not sure if it affects breeding rate)

"The two don't seem to like each other." - still breed, but very slow - different species, same OT

"The two prefer to play with other Pokémon than each other." - won't breed at all - different egg group / same gender / genderless / undiscovered egg group

"The two play with each other, but only as brother and sister." - won't breed at all - siblings

 

Event breeding:
When breeding events, you mostly get eggs of the counterpart species, and very little event eggs. The only exceptions are Mr./Van Bagon, Shadow Mewtwo, and Diancie (Emera), all of which give 100% event eggs (when breeding with Ditto).

 

Breeding Mr. Bagon mostly produces Mr. Bagon eggs, but it also produces some Van Bagon eggs. (Similar thing when breeding Van Bagon; you get some Mr. Bagon eggs)

Dace/Merrick/Allotrope have an extremely low event rate for some reason. I bred 200+ Carbink without getting a single Dace egg. There's only 1 shiny Allotrope on site, and 0 of the other two.

Baby Ducklett, and the event legendaries (excluding Diancie (Emera) and Shadow Mewtwo) cannot breed.

event + event, different OT - best event rate, best breeding rate ("match perfectly")
event + event, same OT - best event rate, second best breeding rate ("very well")
event + counterpart, different OT - second best breeding rate ("very well")
event + counterpart, same OT - breeding rate "get along"
event + ditto, OT doesn't matter - breeding rate "get along"
event + non-counterpart, different OT - breeding rate "get along"
event + non-counterpart, same OT - breeding rate "don't seem to like"

The wiki says that counterpart + event has a better event rate, but I'm not sure whether that's verified by Riako. Since theoretically, the species of the egg should be determined 50:50 by parent:

If they're both events, you'd get X% event eggs and (50-X)% counterpart eggs from each parent, so 2X% event eggs, and (100-2X)% counterpart eggs.

If one is an event, and the other is the counterpart, you'd get X% event and (50-X)% counterpart eggs from the event parent, and 50% counterpart eggs from the counterpart parent, so X% event and (100-X)% counterpart total.
If one is an event, and the other is a non-counterpart species, you'd get X% of the event, (50-X)% of the counterpart, and 50% of the non-counterpart.

Event + event has the best event rate. Any other combination would have an event rate of half of the rate when breeding event + event. Counterpart species shouldn't affect the event rate.

I'm not sure how Ditto breeding works - whether the egg species is 100% based off the other parent, or it's still 50:50 by parent, and if RNG decides it's a Ditto egg, you wouldn't get an egg at all. If the former, the event rate could be either 2X% or X%. If the latter, the event rate would be 2X%, but the number of eggs you get would be halved.

Event + counterpart has a better breeding rate, so it's still better than Ditto.