It depends, as said during one of Elpis quests.
Life forms are separated into two categories: Regular living beings, and arcane entities.
The former have souls, because Nature saw fit to grant them one as they were close enough to "natural" and "sustaining". The latter don't.

Quote Originally Posted by Hermes, Lives Apart
Tell me─do you know the difference between living beings and arcane entities?
It is the presence of a soul. Yet the soul isn't something you can choose to have at will.
No, it manifests only in those beings whose forms adhere to the laws of creation. That can endure on their own.
Beings that do not fulfill this requirement, such as those spontaneously born of magic or natural phenomena, do not have souls.
No matter how much it might resemble flora or fauna, if it lacks a soul, then it is considered an arcane entity.


For the loporrits, the answer is actually in game, on the Triple Triad description. They are arcane entities, and so have no souls.
A race of creatures resembling rabbits and living in Mare Lamentorum. In truth, they are arcane entities assigned by Hydaelyn to crew the star's great ark─the moon.
Familiars having souls isn't unheard of, just uncommon though.


About the ixals, I'd argue that they have souls for a simple reasons: they can be tempered. Tempering is pretty much the Primal messing with your soul's aether balance, i can't see how you'd be able to be tempered without one.


Now clones is a harder question, and one I don't think we have enough information to find a conclusive answer.
On the specific case of Unei and Doga, they had souls. As to how they got their souls, and if it's the same as the original Unei and Doga, we don't know.
Quote Originally Posted by Doga, during The World of Darkness
Know that as long as this light of hope survives in you, our souls will still remain─even if our bodies are lost.
Emet's clones seemed to be simply vessels, pretty much just meat suits to inhabit on a whim.
Quote Originally Posted by Solus, in Seiryu's Wall
I really must thank you for this surplus of vessels. I can mold any host into my own image, but having bodies tailor-made for me in this fashion is so much less tiresome!
It's however completly possible he just didn't even care that they might have a soul in the first place.