The case of Círdan's name is an interesting one, though more from a cultural perspective than from a phonological one.
But let's start with phonology. Phonologically speaking, the [ai] diphthong in Sindarin exists only in final syllables and nowhere else. It arises from [ei], which becomes [ai] in the final syllable but remains the same elsewhere. That is why cair becomes (or rather remains) ceir- in a compound where it is not final.
But here you must be very careful with the order of operations! Note that the primitive form of cair does not have such a diphthong, it is simple ✶kirjā. The diphthong arises later through a process known (in Sindarin phonology) as i-intrusion - that is the stage of keri > keir in the development you listed above - and corresponds with phonological rule S24.
The thing to note here is that the diphthong ei here arises much later than rule OS28 which would turn [ei] into [ī]. To summarise it a bit:
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Primitive [ei] later becomes [ī] - OS 28.
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Primitive [ai] later becomes [ae] - S34.
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[eCi] which becomes [eiC] through i-intrusion (S24), the [ei] later becomes [ai] in the final syllable (S35), but remains [ei] otherwise.
Now, another layer to unfold here is the question of old compounds vs. late compounds (from an in-universe perspective). In the case of "Ship-man", for example, it is the difference between compounding cair+adan (late) vs. compounding ✶kirjā+✶atan and then following the phonological developments into Sindarin (old compound). These two approaches yield different results.
A late compound using cair+adan would indeed yield Ceiradan, as Elaran explained. The compounding process here is a form of reverse-engineering: since [ai] cannot exist in a non-final syllable, it returns to being [ei]; since there are no other issues with the resulting form, no further changes need happen.
An old compound would be very different though. You see, even though ✶kirjā > cair, it cannot actually become ceir- in a compound. The reason for that is because both the stage of kirja > kerja (S8), as well as the stage of keri > keir later, can only happen at the end of the word; as such, if this is the first element of the compound (that is, if it is part of a compound already before becoming cair), then these developments are blocked!
So what actually happens? There are actually two options here, since it's a bit of an irregular case (the application of rule S33, the dissapearance of a short vowel at a morpheme boundary, is irregular with -ja-). The likelier scenario (i.e. the one that is better attested) is that it would become ✶kirja·atan > kiradan (=Ciradan); alternatively it could also become ✶kirja·atan > kiriadan (=Ciriadan).
Now, here's another caveat of phonology and the time when the compound is created:
One of the regular consonant mutations in Sindarin is called "liquid mutation". It may be assumed that at some point, perhaps in late Mannish use, this mutation fell out of use. It is a mutation that would turn [t] into [θ] when immediately following a liquid consonant, i.e. [r] or [l]. So, now looking at "Ship-wright", if the compound is old, then we expect ✶kirja·tanō > Cirdan; if the compound is later but only uses the root √KIR, we may expect Cirthan; and if the compound is very very late (with liquid mutation no longer being applicable) we may expect Ceirdan.
And now lets actually go back to Círdan. You see, Tolkien actually gave us many different forms of his name. Many come from earlier drafts, and may represent conceptual shifts on Tolkien's part; some may also require updating to fit the conceptual changes of how Sindarin phonology works; but it is also possible that, in one way or another, many of them were indeed applicable as epithets of Nowe, the famous Shipwright, in different stages of history.
Attested forms (some updated to fit with the phonology of the later concepts) include Cirdan, Círdan, Ceirdan, and Cirthan; at this point you can perhaps see that all of them are phonologically plausible, in the right context.
All of them... except Círdan, that is...
So why is the vowel long, when it really has no reason to be long? I would suppose that this is representative of a very late form of the name, which actually derives not from cair, but from its plural cîr, then simply compounded with tân. Here, since Sindarin does tolerate superheavy syllables to an extent (if they are stressed, which in this case it is), the vowel then remains long. At any rate, this abundance of attested forms of the name, forms which seem to represent in-universe variations that arose through the millennia, is why I think this is an interesting case from a cultural perspective.
Now I hope I didn't confuse you even more with all of this.
