Flights Through Europe

Yes you can book this separately.

At least try it. All they can say is no. I know of now reason that wouldn't work. It's a multi-flight booking.

I think perhaps the concern originated with someone who wanted to fly on to the second destination on his arrival day, and was told not to do it because if the first flight was delayed, he'd lose the money paid for the second ticket. That concern doesn't normally apply when the second flight is a day or more later, as is the case here.

Also, the transaction is likely to be in Euros, although that really does not pose a problem. Let your credit card co know about the transaction.

This will be fine unless Ethiopia has some sort of highly unusual restrictions. People travel round-the-world not setting foot in their home countries for months or years, in general there are no restrictions as long as you are legally entering the countries with proper visas etc. There are some countries you can't easily enter from others (like US to Cuba until recently, or some Middle East countries to Israel). I can't speak to Ethiopia specifically.

Book US to Europe and also buy on a totally separate transaction Europe to Ethiopia. Don't insist on booking it on one ticket - you have days inbetween and different airlines may be better or necessary for the different pieces.

Personally I would first try to book it as a multi-leg trip. In the past (not to Ethiopia) we have found that multi-leg trips will be cheaper than booking a couple of round trips. We have booked as many as five legs in a single trip. But the airline pricing schemes seem to change daily.

Here are a couple of tools to help your search:
http://matrix.itasoftware.com
This is a Google website serving the airlines and does not sell tickets, but is one of the best sources for information.

www.skyscanner.com --
This site is good for European no-frills airlines and one ways, if you decide to arrange your own ticket for the second leg to Ethiopia.
Both the cost, and the flying time, between Europe and Ethiopia are daunting. A multi-destination ticket looks a better bet. But you must use the multi-city search function on these sites or even airline sites to put together a package that will beat the price of buying tickets separately.

If the cost gets to be too high and you are willing to give up the days in Europe, you can fly directly from NA to Addis. There is a daily Ethiopian flight that leaves Dulles in the morning and takes 14 hrs to get to ADD. Theis isn't convenient if coming from the west since you will have to fly in the previous night and then make the connection but is workable for those who live out East. Ethiopian also flies to Toronto and LAX though the LA flight stops in Dublin I think.
I took the direct flight going to Africa about 4 years ago and it was fine.