I'd say 2 and 3 are the most closely related of the group, with rhinoglottophilia being the only thing differentiating the two, with a proto-word of *pla:'ksa:.ʔá.
1 and 4 then seem to be their closest relatives, but I can't quite figure out what the actual proto-word might be. The first syllable most likely has *pl- as an onset cluster for the first syllable, and *-a:.haʔ later on (the glottal stop triggered the development of high tone in languages 2, 3 and 4 before being dropped, remaining only in language 1). The second syllable is stressed, and then that's all I can work out from there. The middle portion could be *-aisk-, with palatalisation in 1 and 4, but metathesis in 2 and 3. Perhaps, *pla(i)s'ka:.haʔ.
14 seems to have undergone some cluster breaking. I would have said resyllabification, in the vein of Slavic, but it maintains the final affricate, so it's most likely just breaking up consonant clusters. The final affricate would most likely be a retention, either completely or a shifted version of some older sound. Given the final low tone in language 6 it was also likely still a voiced sound. The aspirated velar in 14, though, would appear to be a stress-related innovation.
I get the feeling that language 6 shows signs of earlier metathesis (which could further explain what's going on in 14), since an open syllable which a voiceless onset followed by an another voiceless onset would usually lead to a high tone rather than a low tone. This suggests to me that pre-language 6 had *a.pàl'ka:.hàg.
I'd take a guess at a proto-word of *a.pal'ka:.hag for this group (the long *a: eventually rising and shortening, similar to the development of Old Norse <á> in Swedish).
Of these, 5 and 7 seem the most closely related, the only difference being the non-syllabic /o̯/ in 7. The /h/ in 5 seems most likely to be a retention, so proto-5/7 might have been something like *a.tɕà'ka.há, with the *h dropping out, causing hiatus.
However, two similar vowels in hiatus very rarely do anything but create a phonetic long vowel, not diphthongise the first vowel. Given the form in language 8, this suggests that the stress vowel became a diphthong before the *h dropped out, was maintained in hiatus in language 7 but was reanalysed in language 8.
The tone of the final syllable in languages 5 and 7, and the final consonant of 8, points to the likelihood of tone being an innovation rather than a retention in languages 5 and 7, with a proto-language for all 3 languages being *ə.tɕa'ka(:).hák, with two overlapping isoglosses, one covering 5 and 7 which caused the development of tone and the loss of final *k, and another covering 7 and 8 which caused the diphthongisation of the stressed vowel and the loss of *h.
These two seem to be pretty closely related, sharing a number of similar features. /ɣ/ in language 10 seems to be the result of intervocalic voicing. /tsʲ/ can deaffricate (I believe it happened in languages like Mari), and /sʲ/ to simple /s/ doesn't seem unlikely (especially if the /s/ is laminal rather than apical). I'd take a guess at a proto-word of *tsʲà'ka(:).xà (the vowel length might be stress related.
I honestly can't tell where this one fits right now. It looks like an intermediary between 9/10 and 12/13 (and honestly at this point I'm not sure whether 9/10 is more closely related to 5/7/8 or 12/13). And older form of 11 like *pjà'ka.xà doesn't seem too unlikely, so I'll stop there for now.
12 and 13 are definitely more closely related to each other that to anything further afield. The velar affricate in 13 seems to be an innovation, likely a further development of the aspiration in 12 (which I assume is related to stress.
The coda /x/ in 13 is quite interesting. I can't tell if it's a back-projection of the affricate, or if it's some sort of retention. If it is, it's a very, very old retention because no other language seems to have a consonant in that position besides language 1. The word-final *h seems to be related to the development of tone in language 12, suggesting it's a retention in language 13. The /i/ in the first syllable could be there to break of the cluster.
Stress also seems to have shifted back one syllable, causing aspiration to become phonemic on this group. I'd take a guess at something like *'pʎa:x.kʰa:.xah being the immediate ancestral form, with an older *pʎa:x'kʰa(:).xah before the shift in stress.
This one's a bit tricky, given the disparity in voicing, but the initial /j/ in 16 seems to be an example of glide prothesis, with the resulting /j/ fortifying to /dʑ/ in language 17, suggesting only the /i/ was original. The proto-word likely ended in *-rɑ'kɑ (vowels lengthening in open syllables in language 17, and a stressed *a resulting in /æo̯/ in language 15. I can't tell though whether the long vowel is original or not. There's an entire syllable missing, which would normally yield a long vowel, but the long vowel and the diphthong also appear only in the stressed vowels, and normally, as far as I can tell, long vowels in stressed open syllables remain long, but it's short in 16.
I get the feeling the origin of /f/, /β/ and /b/ might be in *p, undergoing frication in language 15, voicing in 16, and then fortifying to /b/ in 17. This might give the proto-word *ip.rɑ'kɑ(:), also showing signs of reanalysis of the syllable boundary (although it's also possibly that the reanalysis took place everywhere else except this branch, and that *p was always a coda).
So far that gives us:
1/2/3/4: *plais'ka:.haʔ. (northern)
6/14: *a.pal'ka:.hag (central)
15/16: *ip.rɑ'kɑ(:) (southern)
Okay, so, 15/16 seems to be something completely different from the rest, the approximant is *r rather than a lateral, it's missing an entire syllable and the syllable structure as a whole is different. If it is completely different, though, that points to there being an initial vowel which was dropped in a number of branches, rather than being something that popped up here and there.
There's got to be a sort of major eastern group comprising 5/7/8, 9/10, 11 and 12/13, but the exact groups above that are really bugging me, especially because of a) the retention of the initial vowel in 5/7/8, b) the difference in tone between 5/7/8 and the rest of the group, and c) the order in which palatalisation arose.
I get the feeling that 9 might be most closely related to 10/11, giving *pʲà'ka.xà as the proto-word, with an older *pja'ka.xaʔ, where the glottal stop caused the development of low tone (as opposed to the high tone that it causes in the north-west).
If that's the case, then 5/7/8 might have an older proto-word in *ə.tsʲa'ka(:).hak, and the palatal lateral might be common to all of the eastern languages, being retained only in 12/13, but shifting to /j/ independently in 5/7/8 and 9/10/11 (this pre-proto-5/7/8 would be *ə.pʲa'ka(:).hak).
This would give a proto-word for 5/7/8/9/10/11/12/13 (eastern) of *ə.pʎaC'ka(:).hak. (with the division going 5/7/8 vs. 9/10/11/12/13 > 9/10/11 vs. 12/13). The C is whatever the coda /h/ in language 13 comes from (assuming it's a retention at all, but for now I'll assume it is).
So we've got:
Spontaneous palatalisation of post-plosive /l/ can happen (IIRC, it happened in the development of a number of Romance languages), so I think we can assume the the approximant was *l. There was most likely an initial unstressed vowel that was lost in a number of branches and groups independently (IE). Final plosives going from voiceless to voiced is less common in the other direction, so the final consonant was most likely *g, devoicing in Northern and Eastern independently, and further becoming a glottal stop in Northern.
I think the loss of the final syllable in the Southern branch went something along the lines of *-a.hag > *-a.ha > *-a.a > *ɑ:.
So far that gives is *ə.plaX'ka:hag
The X is whatever the explanation for sibilant in the Northern branch and the coda *x is in language 13. There's a chance is was always an *s, became an *h in some proto-not-Northern branch, was lost in Central and Southern, retained in Eastern and survived only in one language.
Pre-aspiration does occur as a result of aspiration, IIRC, in Scottish Gaelic, so *C might not be a feature of Proto-Eastern at all. That leaves *X solely as a means of explaining *s in Proto-Northern. Again, this could have been *s in the proto-world, becoming *h and dropping out in every other language, which does seem somewhat more reasonable. It could have been *h as well if Ixals was right about *h > *s being a change that could happen.
For the moment I'm going to go with *a.plas'ka:.hag, or *a.plah'ka:.hag (I get the feeling the initial vowel became *ə in a couple of areas before dropping out or becoming *i, rather than originally being *a. I'm also still not sure vowel length is an original feature, but for now I'll mark it anyway).