I will try to keep this post short but to see how they looked like etc I have tried to gather what is in my own opinion the best resource for these forms video-wise. So for the first question just look at the clips I have embedded below:-)
Are they still practised in Kukki-Taekwondo? According to the Kukkiwon themselves they are seen as suplemental forms. You can practise them but they are not officially in the syllabus. The Taegeuk forms replaced them and made them outdated. Some schools chooses to teach them anyway as a supplemental forms set so they are still in use but they are not "grading forms".
Why did we replace Palgwe with Taegeuk? This was a political motivated move. In 1965-67 all the schools together made Palgwe forms so they could all practise one common set of forms. At this time the Ji Do Kwan and Moo Duk Kwan which were very big and succsefull were not part of the Korean Taekwondo Association. Later these two schools also joined the organisation and to keep the spirit of all the schools contributing to the forms in that organisation they made the Taegeuk forms and a new Koryo Poomsae with representatives from all the schools including Ji Do Kwan and Moo Duk Kwan. Another reason that is often stated is that the Palgwe forms looked too much like "Karate Kata" while the Taegeuk forms were made to make them look less like Karate forms.
The philosophical meanings of Palgwe forms correspond directly with the Taegeuk forms. So Palgwe Il (1) Jang would have the same symbolism etc as Taegeuk Il (1) Jang and so on.
Palgwe Il (1) Jang (Version one/ Kukkiwon`s version):
Note that there are two versions. The above link shows the latest/last video from the Kukkiwon displaying these forms and use "Momtong An Makki" or inward middle block while the oldest video I can find (and a lot of newer ones) make use of "Momtong An Palmok bakkat Makki" (Outward middle section block using the thumb side forearm).
I do not know which is "correct" or "original" as that is up for debate. Kukkiwon made the DVD where the first clip is from in the link so that is obviously Kukkiwon`s version but the other clip demonstrates what is possibly the older version of the form. If you practise Kick Block Punch Applications you will see that it does not matter as both techniques has essentually the same function but if you look deeper in any way it will matter imensily which version you choose. Personally if I were to import these into my own training (I do not practise these) I would go with the second version as I would use them as additional forms and movement education instead of deep study of applications. Applications would be part of my usage but the primary aim for me would be movement education and as such the outward middle block rarely shows up in Kukkiwon Poomsae making version two a better addition to the Poomsae of Kukkiwon than version one. This is only my personal opinion of course and you are free to do as you whish:-D And that is the beauty of Taekwondo.
As far as I know the rest of the forms are trained in the same way all over so I will only post the links to the Kukkiwon made demonstrations as these are of the highest detail and quality.
Palgwae I (2) Jang:
Palgwae Sam (3) Jang:
Palgwae Sa (4) Jang:
Palgwae Oh (5) Jang:
Palgwae Yuk (6) Jang:
Palgwae Chil (7) Jang:
Palgwae Pal (8) Jang:
And there you have it:-) Personally I am studdying Taegeuk Il - Pal Jang, Koryo - Hansoo Poomsae (including original Koryo), Chulgi Chodan Hyung and Bal Wol Hyung + my own teachers creation the Soak Am Ryu forms. Needless to say I have more than enough forms in my training:p So I will not import the Palgwae forms but I do like them and I can undestand if anyone would like to import these forms into their training. They are more varied and challenging than the Taegeuk forms set. They share the same philosophy and were made using the same movement standard so it is not difficult to learn them and use them in your training. Also this is where we as Taekwondoin came from in the sense that this is the fist creation that were made using representatives from all the different martial arts schools in Korea (excluding Ji Do Kwan and Moo Duk Kwan). The imported Hyung from Karate is where we started as Taekwondoin but this is the first creation from almost all the styles in an attempt to make one forms set all the styles could practise. The Chang Hon Ryu which were made by GM Choi Hong Hi was the first forms set made in Korea but this forms set was limited to the millitary for the most part and the earliest forms were made by GM Choi Hong Hi, GM Nam Tae Hi and GM Han Cha Kyo all from Chung Do Kwan (GM Choi Hong Hi had practised the same Karate style as Chung Do Kwan founder Lee Won Kuk had and was an honorary "Kwanjang" of Chung Do Kwan).
Bonus: Original Koryo were made along with the Palgwae forms but it was also replaced in 1972 by the new Koryo. Here is one interpretation of the original Koryo (slightly different versions exists):
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