To me, "either" seems to mean that there's a choice, for example: "You can have either the ice cream or the cookie," meaning not both. So in an example where it does mean both, like yours, I would not include the either. Another option is rewording it to use "neither...nor", like this: "I like neither the cap nor the shirt," though "neither...nor" isn't used as often in everyday speech.

Either can be used as a synonym for 'as well' : I don't like the cap or the shirt, either. I don't like the cap AND I don't like the shirt either ( tampoco). It is rarely used in spoken English - one can use neither....nor: I neither like the cap nor the shirt.

It is okay to say it without "either" in my part of the US (Massachusetts / New England) too. In fact, I think most people in my area would say it without the word "either."

It is fine without 'either'. You can say 'I don't like the cap or the shirt.'

I'm not sure. I think it is better and clearer with "either." I would use it when writing. I don't know whether it is necessary. In ordinary speech I would probably say "I don't like the cap or the shirt," but I would put some emphasis on the word "or:" "I don't like the cap OR the shirt."