The female speaker really separated the two i's of filii, so much that I thought it was another word. Is that a characteristic of the so-called "classical" pronunciation, or is she just trying to help us remember how it's spelled?
Filii is three syllables. If it sounds like there's a full stop between the two last, it's probably just to make it easier to distinguish.
I do believe she inserted a glottal stop between the two i's and also between e and i in mei, that should be avoided.
Three syllables indeed, but it is written fīliī with the macrons. The glottal stop shouldn't be there, for certain.
Definitely not a feature of Latin of any variety apart from the Germanic strains. Glottal stops are almost entirely unknown to Romance languages. Not to be imitated.
It's not irregular or a diphthong. All fourth conjugation (or "-ire") verbs, like "dormire", "venire", "punire", are characterised by having that "-i-" as part of the stem of the verb, running throughout all the imperfective tenses.
Just a suggestion for the course. I find it very disruptive to have the words introduced as plural, before having me them as singular. If the course could improve this, I would be happy. I know it's not the place for report here, but no way to report that kind of things, and maybe some other people will agree with me.
I don't say it's impossible to learn the plurals, but it requires more effort, and it's a less natural (and efficient) way of learning.