There are few questions on notational choice that seem to come up a number of times. It seems that, while some of this might be somewhat context dependent, it might be useful to get a general idea. I will just briefly go through each of them.
(Q1) Suppose we wanted to identify an ordinal by taking some kind of minimum. We would sometime get two choices: (1a) $\min\{x \in \mathrm{Ord}\,|\,\,\mathrm{some\,condition\,involving\,} x\}$. However, it could also be the case that we could (relatively easily) identify an ordinal $\alpha$ such that the above ordinal could equivalently be written as (1b) $\min\{x \in \alpha\,|\,\,\mathrm{some\,condition\,involving\,} x\}$.
My feeling is that both are basically equally good and which is better is largely context dependent?
(Q2) Suppose we want to write a condition like ($a,b,c,d$ are specific ordinals): (2a)$\forall i \in \mathrm{Ord} \, \forall j \in \mathrm{Ord} \,[((a\leq i<b)\wedge (c\leq j<d))\rightarrow (\mathrm{some\,condition\,involving\,} i,j) ]$.
Now suppose we are dealing with a scenario concerning only ordinals.It seems better to me to just write before-hand that the quantifications are meant to be over ordinals. But in that case, we still seem to get two choices: (2b) $\forall i \, \forall j \,[((a\leq i<b)\wedge (c\leq j<d))\rightarrow (\mathrm{some\,condition\,involving\,} i,j) ] \,\,$ (2c) $\forall \, a\leq i<b \, \forall \, c\leq j <d \,[(\mathrm{some\,condition\,involving\,} i,j) ] \,\,$.
Normally I would think that (2c) is better because it gets right to the main condition we are trying to express. But nevertheless, I am not sure. Another thing (that I initially ignored) seems to be that if we have a different combination of quantifier, it may be difficult to write our desired statement in the form (2a) or (2b).
EDIT: Something else occurred to me. Perhaps the best way may be to express it something like: (2d) $\forall \, i \in [a,b) \, \forall \, j \in [c,d) \,[(\mathrm{some\,condition\,involving\,} i,j) ] \,\,$.
I haven't seen interval notation like $[a,b)$ being used for ordinals before. Is it common place enough that one can use it freely without adding any description? END
(Q3) Suppose we have some (not in 'strict' set-theoretic sense) function $f:\mathrm{Ord} \rightarrow \mathrm{Ord}$ (suppose it is strictly increasing). Now it might happen that we are only interested in ordinals which belong to $range(f)$. But often we will not be using an explicit symbol for $f$ and we have (suppose) something like $\beta_i=f(i)$ for all ordinals $i$. So there seem to be two choices in referring to a relevant ordinal in that case.
We can say: Consider an ordinal $x$ such that $x=\beta_i$ for some $i\in \mathrm{Ord}$. The other option seem to be to define $X=\{\, \beta_i \,|\,i\in \mathrm{Ord}\}$. One could give such a definition in the beginning. Then later on, one can simply write $x \in X$. However, we will need to be consistent with the use of symbol $X$.
Now suppose we have some ordinal $\lambda \in X$ (using notation from 3b). If we have $\lambda=\beta_n$ and we need to refer ordinal $\beta_{n+\omega}$ again there seem to be two choices:
(3a) We just explicitly write $\lambda=\beta_n$ as above and then refer to $\beta_{n+\omega}$.
(3b) We "invent" a notation such $\alpha^{+x}$ ($x$ an ordinal). For example, if $\alpha \notin X$ then $\alpha^{+1}=\alpha^+$ could refer to smallest ordinal that is both greater than $\alpha$ and also belongs to $X$. So, in that case, we could just write $\lambda^{+\omega}$ for our required ordinal.
My feeling is that (3b) is a reasonable notation. But, I haven't seen a notation like $\alpha^{+x}$ (with ordinal $x$) being used. Would it considered to be a reasonable notation or whether it might be better to avoid such short-hand in favour of (3a)?
Perhaps the answer to some of the questions are subjective enough (maybe especially Q3) that it might be hard to give a clear-cut answer? But probably, some perspective on (Q1),(Q2) might be beneficial.