The dot is just a dot ... full stop, end of the sentence. Nothing special about it.
Happy Sam would be: سام السعيد (Sám As-sa3eed). Notice how the adjective (سعيد) gets AL and becomes definite. This is called Attributive adjective, that is an adjective describing the status of the noun (here the noun is a proper name, Sam) and so it becomes attached to it, exactly as in English "Happy Sam". Adjectives in Arabic come after the noun in general, and the attributive adjective specifically mimics the noun in everything (number, gender, definition).
However, in the sentence above, the adjective did not follow the noun (Sam). Sam is a proper name and so it is definite on its own without "THE". But the adjective here is سعيد without AL, which means the adjective here is not attributive, but rather predicative. Meaning, it tells an information about Sam at some point in time. For this reason, when such sentences are translated into English, the verb (to be) as to be used (in this case "is"). Arabic does not use the verb (to be) in forming regular sentences, almost like Russian. The relation is obtained by the presence of the definite article AL and some other characters at times.
Most welcome :)
I didn't go thru the Arabic course here myself since I'm a native speaker but to what I see from other people, the course seems not well-organized actually and still needs a lot more to be done (and they are mixing dialects with proper Arabic). But well, hope you get some good experience here after all :)
The course is still in "beta". Some things that I don't like: 1. Once you download the duolingo app, it automatically opens when you try to go to the duolingo website. 2. New letters aren't introduced separately, but are just given in "pick a word" and "match the sound" excercises.
It happens here on Duolingo in various languages that some answer is right but not accepted because the contributors didn't enlist it. However, Sam's here does also look like, for a non-native English speaker, as if you are saying (of Sam) or (something belongs to Sam).
As a rule of thumb, I personally try to avoid such short hands and abbreviations as much as possible when I do exercises here in any language.