Can I say "all days" instead of "every day"? And "all persons have the right to live" instead of "everyone" so as to emphasize in the first case and to refer to every single individual?
All days and all persons are grammatically correct phrases which mean what they would appear to mean. But there are a couple things to consider.
First be careful when using persons instead of people. People is more commonly used as a plural of person, unless, as onomatomaniak wrote, you want to sound like a lawyer.
Also, "all days" can't really be used the same as "every day" because the former specifes a set of days and the latter specifies that a singular event occurs on each day. So, for example, you can say
But you can't say
If you want to emphasize that every day, without fail, you will eat an apple, with no exceptions ever, you can say
All of those phrases add some redundancy to "every day" in order to emphasize that you are meant to take "every day" as a literal statement and not a metaphorical statement.
Sure you can. Be aware, however, that this construction calls attention to itself and, if employed frequently or weakly, may be more likely to register as an error with your audience.
You'll run into your first example in the phrase today of all days, wherein it's employed for emphasis in just the way you intend.
All persons is seen almost exclusively in legal contexts, where persons is often used in place of people. Instead of that, saying all people may allow you to both use extra emphasis and avoid sounding like a lawyer.