Not at all. In my hometown (Hue City), "ca" is used a lot in everyday conversations. It's equivalent to both a cup/glass and a plastic pitcher.
I'm living in HCM City now. It seems that "ca" is just equivalent to a plastic pitcher in Southern dialects. For cup/glass, people prefer using "ly". So if you come to a food/water stall here and order một ca nước (a pitcher of water) and một ly nước (a glass of water), they will give you two different things.
I don't have information about the use of ca in Northern dialects. Maybe it isn't common any more in these dialects.
In the Southern part of Vietnam (at least from South Eastern region and down under), most of us would use "ca" for "pitchers/jugs" which hold a much larger volume of liquid than "ly" ("glasses/cups/mugs"). People in the North would properly use "cốc" which is the equivalent of "ly". So I suggest that "ca" should be replaced by "ly/cốc" if you want to mean "a mug".
Cái is a classifier or a measure word. Cái typically refers to one unit of an inanimate object. Note that there are other classifiers in Vietnamese as well. Classifiers are seldom used in English but do exist.
For example, in English we say:
- "# sheet(s) of paper" instead of "# paper(s)"
- "# ear(s) of corn" instead of "# corn(s)"
So, I guess the best approximation here would be "one [thing of] mug".
Một was pronounced perfectly, he said: [mot̚˧˨ʔ] that word must be pronounced with a glottal stop, because that is how the tone is pronounced. What is happening here is that the glottal stop is somewhat nasal so it can be confounded with an /n/.
No, "cái" is not "the". In Vietnamese we have words which we call "measure words" or "classifiers". They are used to "class" words in categories. "cái" is basically used for unanimated object (in this case "ca"-a mug. You can also use "cái" for, for example: một cái bàn-a table, một cái ghế-a chair, một cái xe-a vehicle, etc. You will also meet other measure words, for example: "con" for animals; "quả" for fruit; "quyển" for books, magazines; "tờ" for a sheet of papers (or things with the same shape), etc. We don't use measure words when we talk about something in general: "Tôi thích mèo"- I like cat. But we do use them when we talk about something specific or when you use number before noun: "Tôi có một con mèo"- I have A cat. And in you question: A/one mug - một cái ca. Hope it's more clear for you now :-)
He sounds fine to my trained ears. You don't have to pronounce every tone perfectly for people to understand you. As long as "CÁI ca" doesn't sound like "CAI ca" (cái vs cai) or anything else, they would definitely know that you're talking about "cái ca". The difference in tones between neighboring words, though slightly off, is enough to distinguish them.