No, as I pointed out, ‘euro’ (and so also ‘euros’) is written with a lowercase e. Since both ‘euro’ and ‘euros’ are plural forms of ‘euro’, ‘eŭroj’ translates to either of those. (Also, ‘eŭro’ is written with a lowercase e.)
You write the word ‘Euro’ with an uppercase letter when it relates to Europe (like a Euro court). The currency is, like pretty much every other currency, written with a lowercase letter ;).
I completely agree. However, that doesn't mean that we can all just write things however we like just because it is common. A very common error amongst native English speakers is using ‘your’ instead of ‘you're’; does that mean that dictionary should change the definition, because it is common usage? No, of course not. The people just err.
No, pruntigi would mean something like ‘to cause/make to lend/borrow’; I am having a hard time coming up with a reasonable sentence where it can be used though.
You can however just use prunti, because the word both means ‘to borrow’ and ‘to lend’, depending on context and the preposition you use: prunti de means ‘borrow from’, while prunti al means ‘lend to’.
It's prunte|don|i: do give something prunte, i.e., lending. Similarly, you have pruntepreni = prunte|pren|i: to take something prunte, i.e., borrowing. This is just to be more specific, lest there be confusion. Often which meaning of prunti you mean can be inferred from context:
Mi pruntas al vi libron. = I lend you a book.
Mi pruntas de vi libron = I borrow a book from you.
In the first sentence you can use pruntedonas and in the second one prunteprenas to be more precise. See also its definition in PIV.