I would write the first word after the period in capital case, as you did in the second sentence you wrote.
The general rule of writing a word in capital case after a period is still valid even if you write a single word and then a period.
In those cases, the periods are used to give an emphasis to the single words. The pause when "reading" a period is longer than the pause used for a comma; for this reason I would not say the periods are like commas, in the examples you wrote.
I don't know if LaTeX is considered a definitive source for mathematics writing style (although it was developed for typesetting math equations), but this link and this one seem to indicate that, yes, a period would be inserted after the equation in the example
We used the equation
x + y = z.
This is the next sentence.
The Wikipedia Manual of Style (Mathematics), which cites several published mathematics style guides, offers this wisdom (emphasis added by me):
Just as in mathematics publications, a sentence which ends with a formula must have a period at the end of the formula. This equally applies to displayed formulae (that is, formulae that take up a line by themselves). Similarly, if the conventional punctuation rules would require a question mark, comma, semicolon, or other punctuation at that place, the formula must have that punctuation at the end.
The reference for this section is Higham, Nicholas J. (1998), Handbook of Writing for the Mathematical Sciences (second ed.), SIAM, ISBN 0-89871-420-6. The notation states that this is the style adopted by "many mathematics journals," so it is probably a safe choice.
Alternatively, you could use
We used Equation 1.
Equation 1.
x + y = z
This is the next sentence.
Best Answer
No, you don't need one. I agree with you that it looks better without one.
Consider UPS whose slogan is "synchronizing the world of commerce". They don't use a period:
The following image has quite a few corporate taglines:
Most don't have periods. The ones that do do so to be more assertive, which is not something your particular tagline has to convey. The only exception is Microsoft's which is two separate clauses.