Maybe the EU doesn't encourage an application from Ukraine because Russia wouldn't like it?

Russian hostility to NATO membership is often mixed together with imagined hostility to the EU membership. There is the former but not the latter. The latest time the lack of hostility to EU accession was confirmed during Russian president's speech to Serbian National Assembly on Oct 20, 2009, where he said: "We do not have and cannot have any hypersensitivity about new members joining the European Union, including Eastern European states." The question should be closed.

In many discussions, Russian big businesses and their consultants quietly hope for some form of a Ukrainian EU entry, because that would mean backdoor entry for the Russian capital which owns a sizable chunk of Ukrainian one (actually, this is really cross-ownership, links go both ways).

Would the Eastern, Russophone half of Ukraine like an appication for EU membership any more than an aplication for NATO membership?

Overall, Ukraine likes EU much better than NATO: see this graph for EU entry and this one for NATO entry. Regional split on EU entry question is available only in Ukrainian, here. In the last figure, the question is "Should Ukraine enter the EU?", yellow bar is Yes and grey No, lines from top to bottom are for East, South, Center, and West.

Yes, of course the West is more enthusiastic and East less so, but the EU entry is often sold as a package with NATO. It also might mean in respondent's mind that a union with Russia and Belarus is excluded by the EU entry. This book (regretfully, only short excepts and only in Russian are available here) states that 50% of population is for "Eastern" foreign vector, 18% for "Western" and 19% for developing on its own, with "Eastern" dynamics rather stable over time. However, these orientations are not exclusive, as 31% of those wishing to see Ukraine in a union with Russia and Belarus won't mind EU entry, and full 43% of those for EU entry would like to see the union.

So, the conclusion there seems to be: if EU entry would be posed as something that does not exclude close cooperation with Russia, it would be rather easy to generate support of population on 50%+ level.    

I mean, seriously, what fraction of the population would be in favour of EU membership, and what fraction of the Duma?

The latest poll (Sep 2008) is here, again only in Russian. The very first bar plot is yes answers over time, which started from 59% in Mar 01, peaked at 73% in June 03, and fell down to 30% in Sep 08. The number opposed didn't move so dramatically, from 19 to 10 to 27% today. Almost half of the population is currently undecided. Between 2003 and today, there were many reasons for cooler attitudes, such as new members' entry which, in full accordance with Russian predictions, worsened the EU-Russia relations (EU was promising that nothing of the sort would happen... I guess Russia by now considers most West as having precious little credibility). And of course, Sep 08 was right after the Russia-Georgia war where EU as a whole did not exactly play an impartial role.