—What is the mothertongue of most people in Eastern Ukraine, i.e. Donetsk, Kharkov, Lugansk.
These 3 regions are mostly Russian-speaking. You visited Odessa and Kiev for example and you should experienced that Odessa is mostly Russian speaking and Kiev is about 50/50. But if you look at Ramil's map the area wich shown as Ukrainian speaking is not so homogeneous either. In Chernigov and Sumy they speak a mix of Ukrainian and Russian. I'd say it's Ukrainan, but with many Russian words instead of Ukrainian. For example I never heard they say "дякую" there ("thank you"), only "спасибо". Also from my experience people in cities and industrial towns there use more Russian words while people in villages use more Ukrainian words. When I came to Ukraine for the first time I almost couldn't understand language spoken at Ukrainian TV but I could easily understad what "Ukrainan speaking" people say around me.
—Can they speak the other language and how hard is it - is it a sensitive issue / big deal?
Russian is understood by 100% of population. Most Ukrainian speakers can also speak it without any noticable accent except using soft Ukrainian "г" sound instead of hard Russian one, but many people in south Russia also speaks with such "г".
Are there people there who actually regard themselves as Russian, like people in Crimea? Or do they regard themselves as Ukrainians with Russian as mothertongue?
Actually there are not much differences between eastern Ukrainans and Russians besides language. For example about 30% of Russians in Russia has Ukrainan heritage. People with Ukrainian surnames call themselves Russians in Russia while Russian-speaking people with Russian surnames in Ukraine call themselves Ukrainan. It's all very complicated and depends on political preferences mostly.
—How can Kiev satisfy the demands of the protesters in Donetsk and is it necessary?
I doubt it. I feel very sorry for protesters in Donetsk, Lugansk and Kharkov, because there are already cruel repressions against them being done and it's only the beginning. For example there are some reports that the guy whom they elected as "people's governor" on some meeting and who was arrested and dragged to Kiev was tortured till death already.
—Can Putin be trusted when he says that Russia will never enter Eastern Ukraine?
I doubt he can be trusted after he said just a couple of weeks ago on press-conference that Russia is not going to annex Crimea.