What you do will be determined by the particular style guide you follow. If you don't follow one, then do whatever makes sense—but keep it consistent.
The Chicago Manual of Style's guidelines (6.13) would have you omit a period unless it is part of a complete sentence that is not part of another sentence. Also, again unless it is a separate and complete sentence, all examples I've seen have the first letter in lowercase—although this is never actually stated as a guideline).
When an entire independent sentence is enclosed in parentheses or square brackets, the period belongs inside the closing parenthesis or bracket. When matter in parentheses or brackets, even a grammatically complete sentence, is included within another sentence, the period belongs outside . . . avoid enclosing more than one complete sentence within another sentence.
Fiorelli insisted on rewriting the paragraph. (His newfound ability to type was both a blessing and a curse.)
Felipe had left an angry message for Isadora on the mantel (she noticed it while glancing in the mirror).
Chicago (6.98) also discusses the use of other punctuation in parentheses:
A question mark, an exclamation point, and closing quotation marks precede a closing parenthesis if they belong to the parenthetical matter; they follow it if they belong to the surrounding sentence. A period precedes the closing parenthesis if the entire sentence is in parentheses; otherwise it follows.
Come on in (quietly, please!) and take a seat.
If parenthesis is Greek for the act of inserting, is it redundant to insert something in parentheses (i.e., in English)?
You might wonder why a period would be treated differently than other punctuation, but this is no less logical than why periods are treated differently with closing quotation marks—and, in fact, consistent with that different treatment.