Generally, nano size is considered for any one dimension less than 100nm but I have also seen many papers describing their materials of nano size even when dimensions are more than 100nm.
"Nano" has become a buzzword. But this is not the first word that has become fashionable in science. Unfortunately, the grantors often do not have the necessary knowledge to properly evaluate projects (just like grant applications almost never contain an exact statement of the problem, since applicants are justly afraid to "open all the cards"). Therefore, they respond to key phrases that should belong to the fashionable discourse at a particular time. That's the way science works ...
Strictly speaking, you are right: only sized less than 100 nm should be considered as nano. But any particulate does not have exactly sized particles. Usually, the mentioned size means a mean one. So, significant part of particle’s population is smaller than 100 nm. Additionally, even among 100 nm sized particles, agglomeration is notably high that than than among fine or submicron sized population. This, there is no problem to include the case mentioned by yours to nano category.
Yes. For example, nano clay is just a layered silicate, which has nano size in thickness and micro size in length. Still, we consider nano clay dispersed in polymer as a nanocomposite. The same can be applied to the nano tubes. If any one dimension falls in nano level, then we call as nanocomposite.