More elaboration re. 4 - The German-English legal dictionaries provide many translations for the term, Dietl Lorenz has "public corporation" "body corporate under public law" "corporation under public law" and "statutory corporation".
You can find a few examples of both "corporate body under public law" and "public corporation" in the English translations of decisions on the website of the Bundesverfassungsgericht.
I think Körperschaft des öffentlichen Rechts is one single legal form and really think you would be fine with any of the terms given in the dictionaries (my Romaine has gone walkabout again, so I can't look those up for you). I prefer "corporate body", but I'll concede that that may be because I, personally, think it sounds more dignified. But if the Land has published a translation of the statute, it would seem appropriate to use that wording, as suggested above. If your text relates to a specific university, you might be able to find the term they prefer somewhere on their English website.
Many public universities in the US describe themselves as "public corporations" in contexts where they also mention being established in their state consitutions, so that sounds like an analagous term there at least. If you are translating into AE, I suppose that might be a factor to consider.
I would want to leave the "legal capacity" in there, because it is in the original text. Does seem a bit superfluous, if KdöR always have legal personality, but like no me bré and wienergriessler, I'd assume the legislature had its reasons. Perhaps they simply felt it added gravitas.
