Articles – Is ‘Via a’ Correct Usage in English?

They're both examples of a broader concept sometimes termed sandhi, basically meaning 'phonological processes occurring to glue words together'. Whether they're both examples of liaison depends on your point of view and on how you want to use that term.

If you use the term liaison very loosely to mean something informal like 'a consonant introduced between two words to join them together', then you could maybe class English a~an as an example of this. You might also class as liaison in English the 'r' sound you get between vowels in e.g. 'law-r-and order', 'Canada-r-in summer'.

But, from a more technical point of view, if you start to try and model the actual phonological processes that are happening in French liaison, the phenomenon has some fundamentally different characteristics in French compared to English. For example:

• in French, liaison is generally far more 'optional' than in English than in the a~an case, though the case of 'intrusive r' is-- though reversed in terms of its sociolinguistic status [that's a fancy linguistic way of saying that liaison "sounds posher" in French than English]-- the process is maybe more similar;
• in French, the phenomenon is clearly much more widespread than just a handful of words;
• but on the other hand, unlike the case of 'intrusive r' in English, in French the choice of consonant is clearly part of the "definition" of the word to some extent and not just defined by a very general criterion such as 'vowel followed by vowel'-- e.g. if a speaker decides to make liaison in "beaucoup occupier", the liaison consonant will never come out as [z]*;
• in French, the liaison consonant can occasionally surface on the end of the word in question even without the following vowel/liaison context;
• in French, the presence/absence of liaison depends on a slightly 'deeper' surrounding syntactic structure than a~an in English.

So, it depends on how specifically you want to compare 'like with like' when applying the term to both languages. For reasons such as those above, I personally would avoid doing so.

[*] Actually, there are speech errors when this occasionally isn't true.

In the first example, Doctor is being used as the name of the person; the doctor is more of a descriptive phrase. It's short for Doctor <his name>.

tend bar is a set phrase, it's a synonym for being a bartender. It's also similar to the way other people describe their work: a mailman could say I deliver mail, a programmer would say I write code, a garbageman would say I collect garbage, and a composer would say I write music. These are all using the noun to refer to the general concept, rather than any specific item, so no article is needed. You would add an article when you need to be specific, e.g. I write the music in TV commercials.