The monsters, upon entering the minigame, get their min attack, max attack, life, def, and prot modified according to which "cycle" (wiki calls it rounds) the minigame is in. Each cycle is made up of 7 "stages" (what the wiki and most people call waves).
HP = HP * Cycle * 0.8
Defense = Defense + ((Cycle - 1) * 3)
Protection = Protection + ((Cycle - 1) * 2)
(Protection will cap at 80 in this minigame.)
Min and max attack are set per species, and there's a formula to have them increase over time that I don't feel like trying to replicate here (as it involves a different multiplier based on the round). If the game is in dev mode, messages would be output showing the round and attack modifications as monsters spawn in (but this was commented out anyways).
Each cycle there's a check on the egg's HP, if it's below 20% the game should warn you.
The game actually sets your party starting on a specific cycle depending on the party's cumulative level (so that stronger players don't start out killing the weakest mobs I assume).
Cycle 1: 1-500
Cycle 2: 501-1500
Cycle 3: 1501-3000
Cycle 4: 3001+
The spawn timer is calculated per stage/wave, most stages are 30 seconds, then the "miniboss" of a cycle/round sets the timer to 60 and the "boss" of the cycle sets the timer to 120 seconds until the next spawn starts. To give you more time to kill the tougher things, etc.
Finally the rewards.
First off, the rewards vary based on which cycle (round) your party started on and how many stages (waves) in total were completed. The "quality" of your reward is...
Cycle 1: 1/3 of your waves completed.
Cycle 2: 1/2 of your waves completed.
Cycle 3: 2/3 of your waves completed.
Cycle 4: your waves completed.
If your waves cleared is at least 21 (regardless of the cycle you started on), you have a flat 1% chance of a Golden Marble.
If the reward quality number (after cycle reduction if appropriate) is at least 14, then the game multiplies it by itself (waves*waves) and then generates a random number up to 20,000. If the reward number is higher, you get a Blue Dragon's Crystal.
The resulting/final number (waves modified by start cycle) is then set as the quality on the Legatus' Gift exploration artifact given to you. Which is a hella'-weird way to grant rewards, but then again this is Mabinogi.
When you hand it into Belita, she calculates the EXP, Exploration EXP, and Gold as such.
EXP: ((Quality * 80) + (Quality * Quality * 2)) * (Random(8) + 51)
Exploration EXP: Quality * (Random(30) + 55)
Gold: ((Quality * 20) + (Quality * Quality * 4)) * (Random(3) + 9)
Also, the gold count uses a hash on the reward item as part of the random factor, perhaps to prevent people from repeatedly farming for the highest gold output through bugs like people used to be able to do with hot air ballooning rewards.
So let's say you went in with a party with a total level of 3001 or higher and completed 4 rounds exactly. Your final reward "quality" would be 28 (the number of waves completed with no penalty).
194,208 - 220,864 EXP
1,540 - 2,352 Exploration EXP
33,264 - 44,352 Gold
Which, in general, matches what the Wiki lists for the waves that have been filled out. The rewards are within a specific range, and even then completing an extra spawn or two in a round will increase your rewards further, so I think a calculator would be better than a list or something if the wiki wants to fill this info out there.
Then again the exploration EXP needs testing in NA due to the Treasure Hunter update anyways.
Also if you went to 21 "rounds", you'd get 804,384 - 1,072,512 gold.
I don't know where the minigame stops, I just think 21 is an acceptable high limit.
100 rounds? 17,766,000 - 23,688,000 gold, but that's literally 700 spawn waves so good luck.
Each round is 5.5 minutes, so that'd take 9.16 hours straight...