Are all hypothetical machine models for calculating runing time of an alogrithm same?
In RAM model, which is very close to our computers, standard arithmetic operations and assignments take same time disregarding values as far as they fit to computer word. This yields good estimate for runtime and apgorithm behaviour. When some operation takes $c$ time, but constant you may provide it in assymptotic analysis or omit it, becase such analysis takes into consideration length of the input as it grows.

For fine grained computation and small inputs it makes sense to check whether $\mathcal O(5n^2)$ is better than $\mathcal O(100n)$. In assymptotic case constants are dominated by input length, that is why there is no difference between 3 and 20000, so using 1 as time unit makes calculations easier.

It is up to analyzer, how fine grained is analysis, but also which model is chosen, differences here are huge. Between RAM model and Turing Machine model difference is polynomial. Between RAM and Real RAM or BSS machine are vast, instead of word-sized integers arbitrary real or respectively rational numbers could be stored with operations on them taking single time unit. This creates enormous computation gap. So the choice of model is crucial and models are nowhere near similar.

Convention is to use smallest plausible model.
If you sort integers in transdichotomous model, the size of numbers doesn't matter, in RAM it does, it takes unit cost if they fit word size or longer if they don't.