How does the human brain store motor skills for long term?
I second the comments on the question that the question-post looks kind of convoluted. But it seems it all boils down to the question:


  How does our brain store motor skills without any apparent long-term potentiation or protein synthesis?


I personally think the question premise, i.e. "LTP and protein synthesis are not involved in motor learning", is incorrect to start with.


Firstly, motor learning has been linked to long-term potentiation in the primary motor cortex (Avinzino et al., 2015);
Secondly, protein expression is involved in motor learning as well. Fos expression has been used for decades as an immunohistochemical protein marker for neural activation. Fos expression in the motor cortex has been linked to motor learning, alongside synaptogenisis processes (Kleim et al., 1996). 


References
- Avinzino et al., Front Behav Neurosci (2015); 9:105
- Kleim et al., J Neurosci (1996); 16(14): 4529-35