Auslandsjahr?

Yes, you usually apply to a particular country. It does not offer any organization of any country. The more exotic your country desire, the less choices you have.
My organization, for example, has the ideology that one can make great experiences in every country and explicitly encourages applicants to consider other countries as the first wish. There is also an approximate capacity for each country; if, for some reason, there are over-average applications for a given country, it may be that a second or even third request can decide whether to make an exchange year at all. Also in my vintage there were some who actually wanted to Chile but were put to Brazil, who actually wanted to New Zealand and then were placed in Australia, who actually wanted to go to Japan but then eventually went to Argentina. It is, however, not so bad that at this time, when one can go back with the country for which one is accepted, one cannot at all be able to withdraw at that time without the cost of the programme. Better to say, participation in the programme begins only when one has been proposed for a country.

Hmmm, Egypt used to offer it, but I don’t think that is an organization.

In principle, student exchange by organization is not at all with own relatives.

There are various possibilities for financing, but maybe you can think about it yourself first, whether you want to do it and then, if necessary, ask it again in a targeted way (or ask questions about it).