Is $3 \ge 1$ or is it just $3 > 1$?

You are right.

Regarding your friend's concern: The statement 'I am a person or I am a horse' is true. I am always a person, so this is true. It does not matter that it is impossible for me to be a horse.

Similarly, '3 is greater than or equal to 1' is always true. It is always true that 3 is greater than 1. It is never equal to 1, but this does not matter.

This question, like many others, can be resolved by returning to the definitions. For a statement of the form 'A or B' to obtain, at least one of A or B must be true. This is necessary and sufficient. As long as one of A or B is true, the status of the other doesn't matter. It doesn't even matter if the other can never be true. All that matters is that at least one of them is true.

For a more formal approach, consult any textbook on logic.

You are correct, and so is your reasoning: $3\ge 1$ is precisely equivalent to the compound statement

$$3>1\quad\text{or}\quad3=1\;.$$

You might try pointing out to your friends that in order to be consistent, they would have to argue that ‘$4$ is even or odd’ is false, because $4$ is never odd!

The statement "two is even or pigs fly" is true, as is anything of the form "[true] or [false]".

So yes, given two numbers $a,b$, exactly one of the following is true:

$$a<b$$ $$a=b$$ $$a>b$$

But it doesn't hurt anything to combine the true one with a false one, using "or".

Your argument is correct. If $x$ is greater or equal to $y$, this means that there exists a non-negative number $a$ such that: $$y + a = x$$ this of course true for $ 3 \text{ and } 1$ $(a=2)$, so indeed $3 \ge 1$.

In programming, if one writes

if (condition1 || condition2)

where || means OR and condition1 is true, condition2 is, in most languages if not all, not even evaluated - for the very same reason: it does no longer matter. The whole expression is true.