hi there Alberto! :-)
"Although this exercise may seem boring, it is useful"
"Although it's cold, we're still going rowing"
"Even if it rains, we won't cancel the match"
all three are correct and what you would normally say.
"Boring though this exercise may seem, it is useful".
"Cold though it may be, we are still going rowing".
"Rain though/as it may, the match won't be cancelled".
these are correct too, but formal and old-fashioned. You can speak like this, but not really in everyday speech.
Particularly, I'd like to know:
- whether I can use "though" or "as" in every case
this is a bit complicated to explain here.
- why isn't there a verb after "may" in the last sentence? And how can it be used this structure in this way?
Because you are basically saying the same sentence in a different way.
You could say:
"It may rain, but the match won't be cancelled."
but instead this says
"Rain thought it may, the match won't be cancelled."
it is the same sense in both sentences.
:-)