Can inner electrons get excited? Can an already excited electron get excited again without first dropping to a lower energy level?

yes they can. However, to knock an inner electron up and into a higher energy level requires more energy, the deeper you go into the orbital structure. For example, for a low-lying orbital in something like aluminum, it takes ~tens of kiloelectron volts to do the trick, and when the electron decays back to its original state it will throw off an x-ray photon.

Absolutely the ejection of inner shell electrons is possible! As a chemist, this is one of the best techniques for the identification of atomic compositions of materials in a process called X-ray fluorescence (XRF) spectroscopy. Of course, electrons don't have to be completely ejected either. It is perfectly possible to promote them to higher shells, only to see a cascade of other processes happen afterwards to refill the "hole" left behind. Here is a link to a site that gives a very basic overview, as well as some very nice diagrams and examples: https://web.archive.org/web/20131228123922/http://www.amptek.com/pdf/characteristic_xrays.pdf