Ok, so I can't send any of my extra pets to the pound. I can only click the "Pedigree" button. They have no effects, so I don't understand why I can't send them away to make room in my stables so I can breed a Symurgh. Please help.

What pets exactly do you try to pound?

Are you trying to send the pets from your "To the Pound" group? The two Rocs are still babies and need to grow up before you can pound them and the adult in there is curretnly Exploring an Island (check his status underneath his name). You first need to go to Explore while he is active and finish the exploration with him before you can pound him.

I see that now, thanks. But now, I can't breed my Symurghs. I have enough room, it just won't let me breed 'Silver Arrow' with my male

Silver Arrow has reached her breeding limit. You can try to level this pet focusing on strength in order to try getting additional breedings.

wait, you can do that? HOW? And what are the chances?

Strength is the stat that affects the number of breedings. 

Now, if you leave the settings as they are, whenever your pet levels up, the game gives it 6 points which will be randomly attributed to all the 6 stats. So you could end up getting 1 point for each stat or 3 to Strength, 1 to Intelligence, 2 to Agility and 0 to all the others, etc. The chance in this case is the same for each stat to receive the additional point. 

You can however, change your pet's diet. To do this, go to your pet's profile and click on the stars in the Diet Composition box on the left side. If you increase the stars for Animal Fats, your pet will consume more of those each day, but for that when the pet levels, the chance that Strength (the corresponding stat to Animal Fat) gets boosted is much higher. 

To see which food corresponds to which stat, hover over the question mark in the Diet Composition box. It's actually easy though. The top food is for the top stat (Animal Fat -> Strength). The second food corresponds with the second stat (Plant Proteins -> Agility) and so on.

So in your case, you would want Strength at 5 stars so the chance for a Strength gain per level is the highest possible. Don't forget though, in order to raise stars in one food group, you first have to decrease the stars for a different food. 

The exact chance-ratios I don't know though, sorry.