Translation:When are additional tests necessary?
- Notwendig - Not (distress) + wend[en] (turn around) + ig (turn into adjective). It's not really obvious, but if something is used to turn around the distress, it means it's necessary for life.
- zusätzlich - zusetz[en] (to add) + lich (turn into adjective). zusetzen itself could by analyzed as zu (to) + setzen (to set), so adding is just setting something nearby ("to") something else
- Untersuchungen - Unter (under) + such[en] (to search) + ung (turn into noun, like "-ing") + en (plural form). So Untersuchungen = *undersearchings. Not particulary nice, but compare with "understand". If you stand under, then you understand, so if you search under, then you examine.
They are becoming more synonymous in everyday use and here "nötig" would be fine too. However there are definitely cases where they aren't interchangeable. For example, if we had a sentence about someone being in need of something (or needing), the German translation would be "nötig haben" and never "notwendig haben". Think of the difference in English. "When are more tests needed" is pretty similar to "When are more tests necessary", but you'd still say "I am in need of food" and not "I necessitate food".
There is no article. Then the adjective gets strong inflection: http://en.wikipedia.org/wiki/German_adjectives#Strong_inflection in this case it is for the nominative (subject) plural.