Italicized? Depends. You first have to understand what an italic is.
There is a true italic where letterforms are based on handwritten letterforms. There is oblique or fake italic where the roman letterform is slanted. There is a faux italic when a browser needs to render a italic, but a true italic is missing, than the browser can create a faux italic by slanting and scaling the roman.
Does a dot or period (.) get italicized?
True italic: The type designer will design and draw punctuation marks to go well with the italic character shapes. So most of italic punctuation marks in true italic fonts are different from the roman. A: (Most likely) different.
Oblique: If a type designer converts roman letter shapes into a oblique by slanting all shapes, then yes, the dot and all other shapes are slanted. But the designer may choose to alter some glyphs as he sees fit. A: Different, exceptions may occur.
Faux italic: The browsers code responsible for converting the roman into a italic will transform all the characters with the same matrix. A: Always different.
All this matters if you care about typography. At small sizes it's hard to tell the difference between dots. Most people won't ever notice. More people will notice if it goes wrong!
Most roman and italic fonts within the same family have different punctuation shapes. This is because the design principle (concept) of the roman or italic dictates the shape. True italics are based on handwriting. Oblique is based on the roman font and transformation matrix. If a principle is applied than all glyphs will look different.
Yes, design different shapes. Yes, you have the freedom to make exceptions. It's not wrong to copy some roman punctuation into a italic font. When it looks fine, it is fine.
.and upper-case dot
˙? If you're unsure, make a compromise: