I think that it'd be good if there was a link you could click on that would allow you to see a pet's position on the charts, becuase I have no idea how my pets are doing at the moment XD If there's already something like this, could someone please let me know!

Also, is there a section on pets' profiles where you can see how many arena battles they've won?

I'm not too sure about your questions(?), but I'll just answer what I've inferred from it.

I'm not sure what you mean by 'position on the charts', are you talking about the Overall Charts & Weekly Charts? If so, then just move your mouse over to the Community tab, and it's there. JYI, the Weekly Charts only show for that week (from around Sunday/Monday 00:00, server time, to Saturday/Sunday 23:59.) -- As for the Overall Charts, I think it was around the top 20, is only shown.

Now then, as for your second on, err, it's kinda obvious, juct click on your pet, and it should show up on the left-hand side, besides the picture. It's labeled Pet Info 

For the first bit, I mean how do you know if your pet is 100th or 50th or whatever position on any of the weekly and overall charts. Is there a way to find out?

And for the second bit, thanks, I'm not very thorough! XD

There are mainly two ways, both relies on Ctrl + F, then type in either your pet's name or your name.

Oh right, thanks for that!