This is an incomplete answer, and the OP seems to have left MathOverflow several years ago, but since the only answer that has been posted thus far gives an incorrect "answer" to the first question, which has not been stricken out in anyway, I would like to post a reference which **does** answer the first question of the OP --- and perhaps reviving this question will prompt other users who know the literature better than me to point out other sources!

Regarding Q1: contrary to what the earlier answer says (at time of writing), 18.8.1 in Dixmier's book **does not address** cases where $G$ is not second-countable. (There seems to have been confusion arising from the usage, standard at the time, of "separable" to mean "second countable", as discussed in comments to that earlier answer.) However, in the notes to Chapter 18 we find (English translation):

18.9.2. The Plancherel formula can be generalised to non-separable postliminal unimodular locally compact groups. [452]

where [452] is the paper

J. Dixmier, *Traces sur les ${\rm C}^\ast$-algèbres*. Ann. Inst. Fourier 13 (1963) 219–262. NUMDAM link

The precise statement can be found as Théorème 3 (Section 16).

As already mentioned, "non-separable" here means "not 2nd countable"; postliminal (*postliminaire*) means GCR in the sense of Kaplansky. Note that for ${\rm C}^*$-algebras, the equivalence of GCR with Type I (Glimm's theorem) relies on a separability assumption for at least one direction.

A quick look at Dixmier's paper shows that his method is to consider the Hilbert algebra $L^1(G)\cap L^2(G)$ and the induced (Plancherel) weight on the von Neumann algebra generated by this Hilbert algebra; this weight is tracial and its restriction to the group ${\rm C}^*$-algebra $A$ is a densely-defined lower-semicontinuous trace on $A$; he then appeals to earlier results in his paper concerning such traces, where the GCR assumption on $A$ is used to bypass the disintegration-of-von-Neumann-algebras approach of Segal and others (which did require $L^2(G)$ to be separable).

Regarding Q2: The reliance on traces may explain, to some extent, why this approach cannot help with non-unimodular groups. I admit I've not looked up how one bypasses non-unimodularity for 2nd-countable Type I groups; it might be relevant that ${\rm C}^*$-algebras which do not support any densely-defined faithful tracial state can have WOT closures that support faithful normal semifinite traces in the sense of von Neumann algebras. The real $ax+b$ group is an example of such a phenomenon, and the Plancherel theorem for that group can actually be found in the Folland book which the OP mentions.

Regarding Q3: I suspect Kirillov is alluding to decompositions of the left regular representation of $G$ as a direct integral of **factor representations**, which can be seen as a weakened subtitute for a decomposition as a direct integral of irreducible representations. I think some Lie group cases that are not Type I were worked out by various authors in the 1970s/1980s but I am not familiar with the literature. Folland's book (the one mentioned in the original question) has some commentary on this topic but no proofs.

**Coda:** I am a bit surprised to see that Dixmier's paper has relatively few citations, even though it seems to have been written at the same time as several other "foundational" papers of his. Perhaps this reflects the fact that only the second-countable case (a.k.a. separable Type I ${\rm C}^*$-algebras) made it into his book, and it is the book which has become a standard reference rather than the series of papers that went into his book.

To be honest: the lack of citations, even in papers which make explicit reference to Plancherel-type theorems for non-2nd-countable groups that are "close to abelian or compact", does make me slightly concerned that there may be problems with Théorème 3 in Dixmier's paper, perhaps known to experts but not recorded in the literature. Of course I hope that this is not the case!