------ Strunk and White - rule 4
In their rule 4, Strunk and White (and I believe the OP's question too) refer to complex compound structures only, and only those that follow pattern such as this:
[independent clause 1] , [conjunction] (,) [dependent clause to independent clause 2 - introductory clause] , [independent clause 2] .
As do many other authoritative source, Strunk and White state that the second comma (the one in parentheses) should not be used. And for a good reason, too.
Not every pair of commas indicates that the element they surround is non-essential to the sentence or parenthetic. The first comma in the example above separates two independent clauses, and the second one indicates the boundary of a dependent clause that precedes its main (independent) clause. The rule that governs these commas is rule 4: "Place a comma before and or but introducing an independent clause" (the first comma in the example). The commas Strunk and White discuss in this rule are not parenthetic commas (and there are very few dependent clauses that are parenthetic, anyway). However, this advice is valid, widely followed, and is equally correct for parenthetic elements too (as I'll explain below). It is very rare indeed that you'd have a comma immediately following a coordinating conjunction such as and or but.
This rule is for complex compound sentences such as outlined above, where two independent clauses are joined by a coordinating conjunctions "and" or "but" (and a few other conjunctions that are outlined later on in the rule) and where the conjunction is followed by a dependent clause, or an introductory phrase requiring to be set off by a comma, which precedes the second independent clause.
Unlike some answers here suggest, adverbs are a different case altogether.
------ Strunk and White - rule 3
This rule refers to any parenthetic expressions that need to be set off by commas, when a conjunction precedes the parenthetic expression. (Not every parenthetic expression needs to be set off by commas or set off all.) The rule applies to constructions such as but not limited to:
[Subject] [predicate 1] , [coordinating conjunction] (comma here would not be grammatically correct) [parenthetic expression] , [predicate 2] .
She said her farewells, and despite the earlier disagreement, gave me a hug. (Matches the pattern Strunk and White use as an example.)
He would stay for the main course, and if it would please his mother, dessert. (The pattern here is different, but the same rule still applies.)
If it were not for the parenthetic expressions, the commas before the conjunction in the examples above would constitute a grammatical blunder (except when required for clarity, which in those examples they are not) because in the first sentence it'd split a compound predicate, and in the second, compound object. And yet according to the rule, when an expression is parenthetic, making it look like one (with commas on each end) should be so low on the priority list that even the compound predicate and compound object can be made to adjust split, just as long as we avoid a comma after the conjunction - avoiding a comma after the conjunction is THAT important.
So, even when the expression is parenthetic (i.e. not essential to the meaning of the sentence), if the commas are necessary to set it off, the commas don't actually need to look parenthetic - the conjunction is also enclosed within the commas: the commas are for clarity and not to tell the reader to skip what's between them if the reader feels so inclined. No-one who reads anything such as examples above pauses to think: "Hang on! Surely if I remove what's between the commas the sentence won't make sense!" When I read such sentences, the first of the commas goes unnoticed altogether, and the second one is where a natural pause would be, and if it aids clarity, the second comma is very helpful.
A writer may suggest that not every expression following a conjunction is parenthetic (or non-essential), and some could be essential to the meaning of the sentence. When you need to set off an essential expression with a pair of commas, I would argue that the same rule still applies. If we're not concerned with making parenthetic expressions look as such, why would we do that to non-parenthetic ones?
- He always brought me dinner (,) and on Sundays (,) stayed to cook it himself.
- He worked every day (,) and if necessary (,) stayed the night too.
- The children played noisily (,) and when asked to stop (,) didn't pay any attention to any such requests.
- He always moved around the office at a leisurely pace, and only when his phone sounded that special ringtone, ran towards his desk.
The "on Sundays," "if necessary," and "when asked to stop" phrases are all essential to the meaning of the sentences 1-3, as is the clause "when his phone sounded that special ringtone" to sentence 4. Not all introductory elements need to be set of by commas, and I don't see any clarity problems problem with examples 1-3 not using any commas at all (though there may be reasons other than clarity to add them), but the example 4 definitely needs something that would help the reader.
If a non-parenthetic introductory element that needs to be set off by commas follows a conjunction, it's logical to assume that the same rule applies because, if we're not concerned about making a parenthetic expression look like one, then making a non-parenthetic expression look like a parenthetic element is just illogical, if not downright bonkers. But do we need the first comma, before the conjunction or do we only need the second comma? Yes, we need both: try the sentences with the second comma only, and you'll quickly see why you need the first one too.
How will the reader tell the difference better what's essential and what's not in the sentence? I think the reader is smarter than we may assume, and I don't think we need to worry about him misunderstanding the meaning. If there's a risk of misunderstanding, I think the problem is with the composition or the sentence structure.
------ Comma After Conjunction which Begins a Sentence and is Followed by an Introductory Expression
This is a very sparsely covered subject. Most sources will state that only in very rare cases a comma after conjunction and before the introductory expression is needed, but where exactly - no-one seems to be sure.
Everywhere says that such a comma is only justifiable in very rare cases. In my writing, I very rarely feel that a comma is necessary after a sentence-beginning conjunction and before the introductory expression that follows it. Most of the time, I don't see that such a comma adds anything to the clarity or alters the meaning of the sentence at all. I searched and searched for guidance on this, and this is the only source that I could find that makes any sense: "Include commas after coordinating conjunctions that start sentences only when a nonessential phrase or a parenthetical follows the conjunction." (http://theeditorsblog.net/2015/08/27/introduce-me-with-a-comma/) I can see why the two examples provided by the blog author call for such a comma.
"Or, she wanted to know, had I left my husband?
"Yet, and this is crucial, I’d forgotten to pack my pistol."
The author says it's "a nonessential phrase or a parenthetical", but there's also clarity: remove the first comma, and the reader may need to go over the sentence more than once to understand what you're trying to say.
At the moment, unless clarity is at stake I choose to never put a comma after a sentence-starting conjunction followed by an introductory or a parenthetical expression. In my reading of novels by authors whom I love and trust, I have not seen such commas at all. (But I will continue looking.)
------ The Impression I Get
From what I see in editors' blogs and style guides, the general consensus seems to be that a comma after a conjunction makes for inelegant writing and is spurious. Some say that such commas were acceptable before but are now viewed as pedantic. I read somewhere that, although such practice was religiously adhered to many years ago and by many classical writers too, in current climate of trend toward minimal punctuation it makes for a clumsy-looking sentences, mashed your reader stumble over too many commas that don't add anything, and makes the writer look ignorant. And I agree. And just go over Lynne Truss's "Eats Shoots and Leaves" or Caroline Taggard's "My Grammar and I (Or Should That Be 'Me'?)" or Steven King's "On Writing" - anybody who's anybody in writing echoes or quotes Strunk and White. And I don't think that any of us here are in a position where to dispute Strunk and White, CMS, Fowler etc.