What are the top reasons people provide when asked why they want to join the EU or the Russian Federation?

Because Ukraine is fairly large, it has complicated demographics (both ethnically and in terms of opinion).

The educated middle class in Ukraine understands that Ukraine inherited weak (in the sense of huge corruption drag on everything) Soviet institutions, feels a sense of shame because of this, and wants to modernize institutions using western Europe (and e.g. Poland) as an example to follow. I think the main push to join EU is this (obviously people also want to do well economically, but I think pride/shame has a lot to do with this as well). I think Russo-sphere is a profoundly dysfunctional and "third world" place in many respects. For example, here is a (russian-speaking, but pictures speak for themselves) url showing what some hospitals look like in Russia (w/ some comparison to how the elite live): http://lj-editors.livejournal.com/393747.html

The intelligentsia (both in Ukraine and Russia) is ashamed and deeply critical of Putin. Many Russian members of intelligentsia are ashamed to be Russian right now. There are anti-war protesters in Russia that are getting jailed.

It is also true that there is a segment of the population in Ukraine that does not like Russians (and Jews!), and some of these folks are fairly radical, and further some of these folks were involved in the February revolution. Some of these folks view http://en.wikipedia.org/wiki/Stepan_Bandera as a folk hero, although he has a negative status in Russia as a Nazi collaborator (the worst thing you could be in Russia). "Banderan" is a typical slur/catchall used against radical nationalist Ukrainians. These folks have motivations that are obvious.

It is also true that there are a lot of Russians/Russian speaking people living in the Ukraine, and when Ukraine obtained independence it started making moves that made this group unhappy (e.g. mandating Ukrainian for official use, and for education). There is a (racist) perception among some Russians that the Ukrainian language is a kind of "village dialect" of Russian, and not a real language (which is of course preposterous). Some of these folks worry about that an independent, Europe-oriented Ukraine will suppress Russian language and culture, and for this reason may prefer a more Russia-oriented stance.

Further, some (generally older) folks (Russian or not) share Putin's view that the breakup of USSR diminished Russia's place in the world, and for this reason prefer a stronger "USSR sphere" which would include Ukraine staying in Russia's sphere.

How strong is the correlation between the language people speak and their desire to either strengthen the ties with Russia or the EU?

I hope the above answered some of this. I think Putin's invasion showed his true colors, and radicalized Ukrainians against Putin's Russia. For example, in Donetsk (eastern "russian" part of Ukraine), the pro-Ukraine demonstrations are now vastly larger than anti-Ukraine. Odessa had very large pro-Ukraine demonstrations (some say largest in Odessa history).

Do people living in the Crimea mostly listen to Russian media or also Ukrainian and western media?

I do not have the information to answer this. I know independent Russian-speaking news sites are under continuous DDOS attack from Russia (but generally are handling it well, and are staying up). There is a huge activity of pro-kremlin 50-centers ("Kremlinbots" as they are called) on all major news media sites with comments.

The internet (especially social media) is making it far easier to get reasonable news quickly, despite disruption attempts from Russia.

Did at any point, since the crisis started, Russian people in the Crimea seriously feared for their lives/safety?

I would find troops with guns milling around and military tech scary. The point of the invasion is to pass a referendum in Crimea to join Russia under the barrel of a gun. I do not think the Crimeans were genuinely worried about the Russian spetsnaz, e.g. opening fire on them. However, military units in the Crimea who are loyal to Kiev are under tremendous pressure/siege to surrender. Some Ukrainian personnel were wounded with non-lethal stuff the Russians are using (e.g. flash grenades).

How is the relation between Russian people in the Crimea and Crimean Tatars?

What follows is conventional history (e.g. don't need me for this, but providing as a reference):

There is a lot of bad blood between Crimean tatars and russians for historical reasons. The tatars are a turkish people that migrated with the mongols and were a part of the Horde (original connotation of "Yurt"/"Orda", meaning a nomadic tent) that conquered Rus (what we call "mongols", russians call "mongol-tatars".) When the original Horde fell apart, the Crimean tatars had a Khanate that included Crimea and most of southern Ukraine, and were allied with the Ottomans. Tatars were a heavily oppressed minority during Tsarist Russian days, and were relocated from the Crimea to central Asia by Stalin around the time of the second war due to the same worries that led to japanese-american internment.

Despite the bad blood, the sides mostly kept to themselves, and the situation resembled typical ethnic tensions found in many other places (e.g. nowhere close to genocide).