Tali is a more specifically feminine version of the name but I think just as unisex names have become very popular in the US, the same has happened in Israel. So Tali is always female, Tal can be either. There's a similar thing with quite a few Israeli names where there's a feminized form (usually ends in an I or A) and what has basically become a unisex form.
I believe this course has already used Tal as both male and female at different points.
They are definitely distinct names (and I assume have different meanings). Nicknames are big in Israel though (even all the politicians tend to go by nicknames, sometimes even professionally like when they speak in other countries) so I suppose sometimes it could be a nickname but it's also used as a full first name in its own right.
so ברוכה comes from the root 'בָּרוּךְ' (barukh), which means 'blessed', and the root word for 'הבאה' is 'בָּא' (ba), which means 'comes'. So literally speaking, ''ברוכה הבאה" means "blessed is he who comes" (or rather, 'blessed is she who comes', I suppose. What with the phrase being feminine. c: )
No, it's pronounced, but it has a more guttural 'r' sound so it's sometimes difficult to hear... Idk if you've checked forvo but you can hear it a little bit better over there. http://forvo.com/word/%D7%91%D7%A8%D7%95%D7%9B%D7%94/#he
Note that הבא is not a noun, it's a noun phrase - he who comes. But you are actually right. If it were not a set phrase it would be more natural to say הבא ברוך, or even more natural מי שבא ברוך. Not because the adjective comes after the noun but because the sentence's subject typically comes first.
But it is a set phrase. Hebrew isn't strict about word order, and in literary language and set phrases you'll often find irregular word order.
I don't think he is combining the ה's. He's just speaking quickly.
If you didn't speak English very well, and started hanging out with English speakers, you'd probably notice that people were doing the same thing. Native speakers can tell there's a small gap in between words, but it may not be obvious to non-native speakers.
Not sure what you mean about א and ו at the beginning of words - why would you confuse these two letters?
Regarding א and ה at end of words - I doubt if there's a trick, but if you have to guess, guess ה, it's a lot more common. (Especially since a few years ago, when the Academy of Hebrew Language ruled that a ה ending is preferred over the traditional א in most nouns of Aramaic origin. (Example: דוגמה instead of דוגמא. So meta!))