Upd. 08.07.23: wrong use of terminology.
English is mostly an analytic language. Analytic languages are characterised, among others, by constituent word order — specifically, SVO or subject–verb–object order for the English language.
Russian is a synthetic language. It does not have a strict word order, nor does it require sentences to be complete. Incomplete sentences are widely used both in "high" and "vulgar" variants of the Russian language.
The "Господи, помилуй их всех, давешних, сохрани их, несчастных и бурных, и направь" example is different, however. I'd rather not go deep in analysis because this is a complex sentence in an archaic style, and my days of formal syntax analysis are long gone. Basically, this is not the case of an omitted object. The "давешних", "несчастных и бурных" are descriptions of "их всех"; therefore, a simpler and more modern variant of the sentence would be "Господи, помилуй, сохрани и направь всех давешних, несчастных и бурных."
Is a pronoun implied, or not?
It is not about the pronoun. It is about an implicit or explicit object or subject.
Targets and objects may be omitted if they are easy to determine from:
the context, or
the situation (ongoing, IRL), or
Typical example: there is an ongoing examination. A student exists in the classroom. Another student asks them: "- Ну как, cдал(а)?".
In English: "- Well, did you pass?".
(Exams in high schools in Russia are usually held closed doors with a limited number of examinees at a time; everyone else waits outside in the hallway).
This is a simple sentence with an introductory constructions conjunction ("Ну как"). If we drop it, we are left with a definite-personal sentence ("определённо-личное предложение") of a single verb "сдал" (or "сдала" in feminine form). Both object and subject are not present, but they are obvious from both the context and the situation. The shortest full sentence might be "[Ты] сдал [экзамен]?".
There are several types of incomplete (or single-component) sentences. You can read more about it here: https://www.academia.edu/68109854/Classification_of_Incomplete_Sentences_of_the_Russian_Language, or here on Wikipedia (in russian): https://ru.wikipedia.org/wiki/%D0%9E%D0%B4%D0%BD%D0%BE%D1%81%D0%BE%D1%81%D1%82%D0%B0%D0%B2%D0%BD%D1%8B%D0%B5_%D0%BF%D1%80%D0%B5%D0%B4%D0%BB%D0%BE%D0%B6%D0%B5%D0%BD%D0%B8%D1%8F.
If you spoke these sentences fully and correctly in Russian, would you add a pronoun?
For those particular examples, it is a matter of personal preference.
Could you choose either way, to include the pronoun or omit it?
In short, yes.
How would the meaning or the tone change if you included a pronoun?
I struggle to provide a general explanation. Let's say it would be more targeted. For example, "Не плачь, Григорий, ступай к Марфе, она утешит, спать уложит." Without knowing that Grigorij is her husband, one would assume that comforting is what Marfa does; she'll help anyone. Martha is just that nice of a lady. Adding an explicit reference to the object doesn't change intonation at all. The difference is purely cosmetical.
Knowing that those two are married, it's just obvious that Martha would do something like that. It's her man. Why wouldn't she? That's what a good wife does. Having an explicit object makes zero sense. "- What do you mean,
Martha will comfort ME? What? Are you saying that my wife comforts someone else, you bastard?"
That's how I would interpret it if I really, really had to. Because, once again, the intonational change is almost or completely null.
For the second example, "А не убил, так еще приду убить.", first of all, your translation is a bit off. The "a" preposition here doesn't mean "but"; it means "and if". "- Madman! You murdered him!". "- Serves him right! And if [for some reason] I didn't kill him, I'll come back to do it". Alternatively, "- Madman! You murdered him!". "- Hopefully, I did! And if not, I'll come back to finish the job."
An explicit object, "him", makes no sense here. It would only make the whole sentence clumsy. They are obviously talking about that old man. Who else could it be?
Can you omit pronouns everywhere?
No, not everywhere. As mentioned above, it depends on the context or lack of such and situation.
I would, however, advise inexperienced speakers to use a "default" SVO order and complete sentences to avoid unintended intonations. Although one will, of course, sound "robotic", "boring", and "unnatural" (which is expected from the non-native speaker), the risk of being misunderstood, looking stupid, or even insulting someone will be minimised.
Let's take the first sentence of the famous child's poem "Мячик" ("The ball") by Agnia Barto as an example.
The original text uses the SVO order (object is not present because the action is self-oriented):
"Наша Таня громко плачет".
The literal translation will be "Our Tanya loudly cries", which, I believe, is grammatically correct but sounds really theatrical; the "regular" translation would be "Our Tanya is crying loud".
Now, we may put those four words in any order, and the meaning will remain the same. Well, kind of. Intonations, however...
I am not autistic enough to go through all 24 permutations; let's try a few notable ones.
"Громко наша плачет Таня" -- ough, that'll be an epic 200 pages poem, right?
"Громко плачет наша Таня" -- it's gonna be a lengthy description of Tanya's personality flaws, right?
"Наша Таня плачет громко" -- other Tanyas are no match for our Tanya when it comes to crying loudly.
"Плачет наша Таня громко" -- look, I do appreciate our Tanya's vocal abilities, but maybe she could moderate it a bit?
I am, of course, joking quite a bit. But I hope you got the point. Russian language, like all Slavic languages, is extremely flexible. This, however, should be exercised with caution. It makes Russian an easy language, but it also makes it impossible sometimes.