All your examples are different and their acceptability must be taken case by case. But remember that parenthetical information is parenthetical; i.e., it is additional unnecessary information.
I'd say that your first example (best friend Jim) is a perfectly acceptable construction.
In your second example I assume that you intend for "the Leased Premises" to refer to the 600 sqft space, but from the position of the parentheses it's more likely going to be interpreted as referring to "attachment A" so I would not recommend using sentence 2. Also, if this is a legal document and you are trying to establish the link between the two things then the information in the parentheses is no longer optional and thus the sentence should not be formulated that way, Use something like: *hereinafter referred to as The Leased Property.
The third example seems ok, except, again, if you are trying to establish a relationship.
The fourth seems awkward- either his name is "Rattlesnake Mike" in which case it's not really appropriate to refer to him as just Rattlesnake, or he is a Rattlesnake (i.e., a member of the Rattlesnake team) in which case it would be more appropriate to say, "One of the Rattlesnakes (Mike) came by the other day" I suppose if Mike's nickname were actually Rattlesnake, then your sentence would be correct but you'd probably only choose to use the nickname if the person you were talking to would recognize it and therefore the parenthetical would be unnecessary. And if you were talking to a group, some of which knew him as Rattlesnake and some as Mike, you might either decide to only refer to him as Mike or to add slightly more to your parenthetical: Rattlesnake (Mike, for those of you who don't know) came by the other day.
