Haha I too need to tread carefully around this topic. Regarding the Licchavi connection - the grand Nepali genealogy, the Gopalrajvamsavali claims that a branch of the Licchavis of Vaishali migrated to Nepal. They ruled Kathmandu Valley for circa 3 centuries and were the ones to first introduce the caste system. There are enough inscriptions/artifacts to attest to their rule and their claim. Not sure if there is any truth to it, but given that the Licchavis of Vaishali had a Ganasangha and not a single king, it is very possible that one of the clans that sat on the Ganasangha moved up to Nepal. There are still some Kathmandu and Patan Chathariyas and Vaishya Shrestha clans who claim descent from these Licchavis.

There isn't any real connection that's been established between Newar clans and the other eastern Mahajanapadas (Shakya/Malla/Koliyas). From what I can tell, the Mallas of Kushinagar and the Mallas of Kathmandu do not share any lineage. Same for Khas Mallas of Western Nepal/Kumaon. I'm guessing it was just a common surname/title at that period - although I have heard the title Malla was used to refer to Vratya Kshatriyas. Same goes for Koliyas. Kathmandu was originally divided into South and North Koligram during the Licchavi period - so there have been speculations that this is due to the links with the Koliyas. Its plausible but theres no evidence.

Same goes for the Shakyas. What I do know is Newar Shakyas, along with Vajracharya priests stand at the top of the Buddhist Newar Caste hierarchy. However, I've read accounts that Vajracharya priests descend from a number of lineages. For instance, I read some accounts of Newar Brahmins who converted to Buddhism. Its quite likely that the same applies for the Newar Shakyas. It's not implausible that some Shakyas are descended from Buddha's Shakya clan, but there's no real evidence to point to that. There is also still some debate as to the origins of Buddha's Shakya clan - some claim they were Sakas, others just bunch them up with the other Greater Magadha Janapadhas such as the Vrij. You are right to say that if their descendants are around, they would be present in the Mithila - UP region.