1. You have these stats on the patch notes: https://www.lineage2.com/news/dawn-of-heroes-patch-notes
Basically it's: low-grade (gray background, stage 1), mid-grade (green background, stage 2) and top-grade (red background, stage 3). Like @mixa said, the stats are better if the pet is higher grade / stage. You can improve the stats, but a low-grade pet won't have the same stats as a top-grade pet.
2. I believe it's completely korean random. So 1:15 chances of getting any of the pets. But a percentage like @mixa suggested would make sense on NCWest servers
3. No. You cannot upgrade the grade/stage of homunculus. You can increase the homunculus level from 1 to 5, but a low-grade lvl 5 homunculus won't be the same as a mid-grade lvl 5 homunculus and much less a top-grade lvl 5 homunculus. Not the best comparison, but think it like this: a +5 Circlet is not the same as a +5 Noble Circlet and not the same as a +5 Radiant Circlet, even if they are all +5.
And finally, basically yes. Seems like it's korean random (or worst, NCWest random), so you'll have to keep trying until you're happy or until you give up trying keep making as much "Enchant Points" as you can per day and when you get the pet that you want, you can raise it's level, enchant stats, etc. Until then, pray for the korean random luck