I'm in the 3rd mission of Persephone's campaign and I noticed that the enemy hero (or whatever you're called) he only needs to touch the souls to gain them while I have to summon Sac Doctors and then wait for them to return to my Altar. This seems like a very unfair advantage and I don't really know how to combat it.

In this mission you recieve the Grappling Vine spell which is awesome but of course it doesn't work on the enemy hero because he uses some kind of fire shield.

I also wonder how mana is calculated? Is it better to have more manaliths and more mana boars, but how do they work?

When you've killed enemy units, those are red souls that you need to convert with Sac Doctors. When units that you've summoned die, their souls are blue and you can just run over them to pick them up. So red souls that you see are blue for your opponent. He has to convert your souls too. The way to combat this is to specifically target the enemy wizard and kill him before he can collect all his dead souls, that way he has to retreat to regenerate at a manalith or near his manahoars. That will give you time to take his units back to your alter for conversion.

I don't know the formula, but generally the farther you are from your own manaliths, the slower you regenerate mana. More manahoars can help you regenerate mana faster up to a point. I prefer 3 unless it is a huge map where I'm operating far from my own manaliths in which case I max out at 4 or maybe 5 if I have extra souls. More than that doesn't seem to provide any additional benefit. You basically want to have all the manaliths you can since that means your opponent doesn't have them and you can move around quickly by teleporting to them.

There's more information here with a better explanation.
http://sacrifice-shiny.wikia.com/wiki/Manahoar

I actually try to target the enemy wizard but I kind of made a weird move in the beginning. Since I prefer to play defensive until I get the upper hand I got all my units to be Guardians at the first mana lith and I don't think I kill them nor does the enemy wizard have much of a chance to do anything but he has all the other manaliths so I assume he has faster regen than me.

Good to know about the red and blue souls, I hadn't thought about that way but it makes sense. Great that it's equal for both.

Until the last mission, when I'm afraid you'll see exactly the behaviour you describe, but it's not a bug, just very hard.