1. What you have:
Animae celare bestias exiguae laxis vestibus illicitus est.
What you are trying to say:
It is forbidden to conceal beasts of little intellect within the folds of your garment.
What I think it should be:
Celare bestias animi exigui in sinibus vestis tuae illicitus est.
Changes: I moved some words to be closer to the ones they modify. I tried to preserve the word choices you made, since nearly all of them were near in meaning to the sense you are getting at. I changed animae to animi because you want the word animus not anima. The former means "intellect" while the latter refers to the spirit. You could also use mens (mentis in this case) to mean "intellect" as well. I changed laxis to sinibus because laxus means "empty," where as sinus means "fold." Finally, I changed vestibus to vestis because you are saying "of the garment," which requires a genitive. I added tuae after vestis to convey the "your."
2. What you have:
Nam eum hinc ferunt.
What you are trying to say:
For they beareth him hence
What I think it should be:
This one seems fine to me!
3. What you have:
Diabolo se agnosci non placet
What you are trying to say:
The Devil does not like to be recognized.
What I think it should be:
Diabolo agnosci non placet.
Changes: I only removed se because it is a reflexive pronoun in the accusative (i.e. the direct object), but passive verbs usually do not take objects. Literally, this one translates to "It is not pleasing for the devil to be recognized (himself)." You can see why the se (himself) does not seem to fit.
4. What you have:
Nam idem quod eos edit edunt.
What you are trying to say:
For they eateth that which eateth them.
What I think it should be:
Looks fine to me. Just note that it literally translates from Latin as "For they eat the same thing which eats them" (idem means "same"). This slightly different but I think it works well with what you are trying to accomplish.
5. What you have:
Ferens infantes exit.
What you are trying to say:
It came out carrying infants.
What I think it should be:
Ferens infantes emersit.
Changes: Now, you've already received commentary on this one in a separate question, but that's okay, I guess. The reason I prefer emersit over exit is that of connotations. Exit means "to go out/away, to become visible," but it also can mean "to die" or "to sprout." It would probably be pretty clear from context which one you mean, but emersit lacks these other connotations. One can also look at the English derivatives of the two words. Exeo (exit) leads to the word "exit" which usually means "to go away" or "to go out of the current place" which implies a motion away (not always, though). Emergo (emersit), on the other hand, leads to "emerge" which means "to come into view." I think this more accurately captures the picture you are painting. you obviously cannot rely solely on English derivatives when making word choices, but it can be a useful tool sometimes. If you are dead set on using exit, you should put it in the perfect exivit because it is in the past tense.