Rearrangment during alkyne synthesis
Your questions:

Q1) How the rearrangement from (G) to non-1-yne can happen (i.e., its mechanism)?

For example, see the following mechanism suggested in Ref.1 as the Acetylene Zipper reaction:



Q2) Why Sodium Amide in particular yields terminal alkyne, unlike the fused Potassium hydroxides why almost always yield internal alkyne?

To answer your question, I have to dig down literature to the 19th Century. To my knowledge, sodium amide ($\ce{NaNH2}$) is not the best base to use in this purposes. The best base, in particular, is the monopotassium salt of 1,3-diamminopropane or simply, KAPA (see mechanism), a difunctional “superbase.” According to the Ref.1:


  KAPA produces exceptionally rapid migrations of triple bonds from the interior to the terminus of the carbon chain in seconds at $0^{\circ}$.


On the other hands, from a synthetic standpoint, $\ce{KOH}$ as a dehydrohalogenating agent, tends to promote the shift of the triple bond away from the end of a chain, which is sometimes considered as a disadvantage (Ref.2).

Q3) Is there any limitation by how much this alkyne formation site can shift?

There are no limitations. Yet, if one substitution of disubstituted acetylene is branched, the triple bond migrate through the straight chain to the end (Ref.2). Mean time, the reaction has some interesting facts:


An equilibrium between alkynes and allenes can be established using basic catalysts (see the mechanism). For example, when very strong bases such as sodium metal ($\ce{Na}$) or sodium amide ($\ce{NaNH2}$) are used, the rearrangement of a triple bond of a disubstituted acetylene tends to migrate to a terminal position. This isomerization equilibrium shifts toward forward direction in favor of 1-alkyne formation due to the fact that formation of the terminal metal acetylide salt, which is stable, precipitated out from the solution, thus automatically removing itself from the reaction mixture.
Favorskiî was the first to discover that 2-alkyne could be irreversibly removed from solution by isomerization to the 1-alkyne and resultant precipitation of the sodium salt in 1887 (Ref.3). This procedure used metal $\ce{Na}$ as the strong base, and thus, has the disadvantage that some olefin would produce  as a result from hydrogenation of the alkyne by the hydrogen liberated upon acetylide formation. On the other hand, $\ce{NaNH2}$ when used in inert solvents, was shown to accomplished the same rearrangement but avoids the reduction problem (Ref.3,4). Temperatures greater than $\pu{100 ^{\circ}C}$ are employed with $\ce{Na}$ metal while the best results with a $\ce{NaNH2}$ catalyst are obtained at $\pu{150-160 ^{\circ}C}$. The data are summarized in the Table (vide infra). The rates of the rearrangement differ with the structures of the compounds isomerized. In general, 2-alkynes rearrange faster than 3-alkynes, and 1,2-alkadienes (allenes) rearrange faster than 2-alkynes.  The amount of 1-alkyne formed is proportional to the reaction time, and equilibrium is obtained earlier at higher reaction temperatures (Ref.3).
Later studies have found that $\ce{NaNH2}$ brings about the propargylic rearrangement in liquid ammonia at room temperature with essentially quantitative hydrocarbon recovery. (Ref.3). The ratio of 1-alkyne, 1,2-alkadiene, and 2-alkyne varied with the amount of $\ce{NaNH2}$ used. High $\ce{NaNH2}$ concentrations favored the formation of 1-alkynes. Equivalent or excess amount of $\ce{KNH2}$ in liquid ammonia, sometimes in admixture with HMPA, have also been shown to convert 2-alkynes to 1-alkynes (Ref.3,5).
The advent of newer strong bases has made the isomerization to terminal acetylenes a very facile reaction. The addition of dialkylacetylenes to a slightly excess of KAPA (the monopotassium salt of 1,3-diamminopropane) results in extremely rapid, quantitative isomerization of the triple bond to the terminal position where it is trapped as the acetylide ion (Ref.1).


Table:
$$
\begin{array}{ccc}
\hline
\text{Disubstituted alkyne} &amp; \text{Base} &amp; \text{Temperature} &amp; \text{1-Alkyne} &amp; \text{Disubstituted alkyne} &amp; \text{Ref} \\ \hline
\ce{C2H4C#CCH3} &amp; \ce{Na} &amp; \pu{100 ^{\circ}C} &amp;  \pu{4\! -5 h} &amp;  n\text{-}\ce{C3H7C#CH} &amp; 3\\ 
n\text{-}\ce{C3H7C#CCH3} &amp; \ce{Na} &amp; \pu{100 ^{\circ}C} &amp;- &amp;  n\text{-}\ce{C4H9C#CH} &amp; 3\\ 
n\text{-}\ce{C13H27C#CCH3} &amp; \ce{Na} &amp; \pu{200\! -220 ^{\circ}C} &amp;  \pu{15\! -20 h} &amp;  n\text{-}\ce{C14H29C#CH} &amp; 3\\ 
n\text{-}\ce{C4H9C#CCH3} &amp; \ce{NaNH2} &amp; \pu{160 ^{\circ}C} &amp;  \pu{12 h} &amp;  n\text{-}\ce{C5H11C#CH} &amp; 3,4\\ 
n\text{-}\ce{C5H11C#CC5H11} &amp; \ce{NaNH2} &amp; \pu{210 ^{\circ}C} &amp;  \pu{7 h} &amp;  n\text{-}\ce{C10H21C#CH} &amp; 3,5\\ 
n\text{-}\ce{C4H9C#CC2H5} &amp; \ce{NaNH2} &amp; \pu{170 ^{\circ}C} &amp;  \pu{9 h} &amp;  n\text{-}\ce{C6H13C#CH} &amp; 2\\ 
n\text{-}\ce{C6H13C#CC6H13} &amp; \ce{K(NH2C3H6NH)} &amp; \pu{20 ^{\circ}C} &amp;  \pu{1 min} &amp;  n\text{-}\ce{C12H25C#CH} &amp; 1\\ \hline
\end{array}
$$
Entry 4: The catalyst used was finely divided $\ce{NaNH2}$ suspended in mineral oil. 

References:


C. A. Brown, A. Yamashita, “Saline hydrides and superbases in organic reactions. IX. Acetylene zipper. Exceptionally facile contrathermodynamic multipositional isomeriazation of alkynes with potassium 3-aminopropylamide,” J. Am. Chem. Soc. 1975, 97(4), 891-892 (DOI: 10.1021/ja00837a034).
T. L. Jacobs, “The Synthesis of Acetylenes,” Organic Reactions 1949, 5, Chapter 1 (78 pages) (https://doi.org/10.1002/0471264180.or005.01).
Hermon Pines, Wayne M. Stalick, In Base-Catalyzed Reactions of Hydrocarbons and Related Compounds; Academic Press, Inc.: New York, NY, 1977, “Chapter 3: Isomerization of acetylenes and allenes,” pp. 124–204.
H. H. Guest, “Rearrangements of the Triple Bond,” J. Am. Chem. Soc. 1928, 50(6), 1744–1746 (DOI: 10.1021/ja01393a036).
L. Brandsma, In Preparative Acetylenic Chemistry (Studies in Organic Chemistry Series: Vol. 34); 2nd Ed.; Elsevier Science: New York, NY, 1988, “Chapter XI. Base-Promoted Interconversions of Acetylenes,” pp. 231–246.
T. H. Vaughn, “A New Reaction of 1-Iodoacetylenes and Some New Mercury Acetylides,” J. Am. Chem. Soc. 1933, 55(8), 3453–3458 (DOI: 10.1021/ja01335a073).