Should you write university names out in full, or use the common abbreviation?

When the abbreviation is actually better known than the full name, go with the abbreviation. Otherwise, use the full name. That will render the last option you proposed (though it would be fine to write TU Munich as well).

Assuming you're applying for an academic job, I recommend including a bulleted list of full names in your CV, in either alphabetical order or (if you also include dates) chronological order. There is absolutely no reason to be concise in your CV; ink is cheap, especially since nobody uses it any more. (If you really think a list of all talks is too long, cluster by year, but still use full names in each year's one-line list.)

Invited talks

• California Institute of Technology
• Carnegie Mellon University
• Eidgenössische Technische Hochschule Zürich (twice)
• King's College London
• Manchester Metropolitan University
• Massachusetts Institute of Technology
• Technische Universität München
• University of California, Berkeley
• University College London
• University of Southern North Dakota at Hoople (14 times)

If you want to emphasize this list in your cover letter, you can write "I have given invited talks at several leading institutions, including Caltech, ETH Zürich, and TU Munich; see my CV for a complete list."

And yes, you should use the word "invited".