1. Is there know set of operations for which uncomputable functions are, let's name it down-unclosed? I mean a set of operations which takes two ( or more) uncomputable functions and return computable function? It is obvious it depends on certain property, like rates of growth, so are this hypothetical internal operations structured like onion, being internal only inside shells of similar growth? So what is known for other classes of functions? non primitive recursive (NPR)? For primitive recursive answer is known.
2. Or relaxing a bit, it returns an uncomputable ( or NPR) function, which grows much slower?
3. **Is there any general notion describing such relationships ( "down-unclosure" under set of operations in bigger set which has built in order?)** Maybe category abstract nonsense is ready defined?


Example regarding not primitive recursive function:


Let A be Ackerman function, so it is not primitive recursive ( it grows faster than any primitive recursive function).
Obviously A + A is another NPR function, equivalent to original one, and similar for multiplication or exponents etc.
But $$A/A$$ looks like quite normal computable function, an probably there are variants like linear combinations fraction or maybe even complicated functional composition on arguments ( for example $$A/(A°g)$$ where g is linear function and ° means function composition, so I mean Ackerman fraction of two functions, one of them with argument shifted $$x \to ax+b$$).


Down-unclosure:


Example of down-unclosure is visible in various areas. There are a proof techniques based on lowering some parameters in the bounded from below set ( for example in natural numbers, contraction of volumes, or more generally measure of sets etc ), various analytical techniques based on control of remainder growth, or even set theoretic techniques based on "almost all" elements which may be seen as certain types of such "down-unclosure" trick.



Internal operations on uncomputable functions
The jump inversion theorem (Friedburg 1957) shows that any Turing degree $d$ above the halting problem is the jump of another degree $d=b'$, which means that $d$ is Turing equivalent to the halting problem relative to $b$. 


Friedburg's original construction can be viewed as a specific map from any sufficiently non-computable set $d$ to a strictly simpler set $b$, with the property that $d$ is Turing equivalent to the halting problem relative to $b$. 


There are numerous generalizations and refinements of the jump inversion phenomenon in computability theory.


