Latin works like Slavic languages in this matter. There are no articles because cases are used to denote the meaning.
That is not entirely true. In English, the indefinite article "a/an" can emphasize the fact that there is only a single of whatever, but it can also mean that a one particular noun is specified.
"unus" would tend to be used in Latin to specify one out a particular group.
Dormit can be either 'sleeps', 'is sleeping' or 'does sleep'. It depends on context.
An equivalent to the definite article "the" does not exist in Latin. When we translate to English (which requires the use of a definite or indefinite article ["a"/"an"] in most cases) we have to add "the" or "a" (in the singular) otherwise it would not be grammatically correct (Girl is sleeping does not work).
The "is" appears when you translate from the Latin present tense to the English progressive present tense. This is allowed since the Latin present tense does not differentiate between the English present progressive (The girl is sleeping.) and the simple present (The girl sleeps.). Of course, there are some cases where you will be only able to translate to one of those two English variations. In short, dormit = 'he/she/it sleeps' and 'he/she/it is sleeping.'