Are there parts of Ukraine that genuinely prefer to be part of Russia as opposed to Ukraine?

Yes. The Donetsk and Luhansk separatist republics were created by pro-Russian Ukranians with Moscow's backing. Russia officially annexed them in 2022.

It's all a bit conplicated though. You have ethnic Russians who consider themselves Russian despite being born within the borders of what we known as Ukraine. You also have many who were born in Soviet Ukraine prior to the collapse of the USSR in 1991 who still feel an affinity for Russia. And pro-Russian politics have existed for as long as independent Ukraine has.

My guess is that even if the war ended today and the borders returned to the pre-2014 borders, there would still be a pro-Russian insurgency in eastern Ukraine for a very long time.

That is indeed the case, it's important to keep in mind that Ukraine was created by USSR out of parts of Russia, Poland, Hungary, and Romania. I highly recommend watching a lecture that Mearsheimer gave back in 2015 to get a bit of background on the subject. Mearsheimer is certainly not pro Russian in any sense, and he gives an objective analysis of the situation. Let's take a look at some slides from the lecture here. First, here's the demographic breakdown of Ukraine:

here's how the election in 2004 went:

this is the 2010 election:

As we can clearly see from the voting patterns in both elections, the country is divided exactly across the current line of conflict. Furthermore, a survey conducted in 2015 further shows that there is a sharp division between people of eastern and western Ukraine on which economic bloc they would rather belong to:

If there are parts, they can move. What a few people want doesn't mean national borders should be taken by force by another state.

Perhaps before the 2022 invasion. You can see for example the first round of the last presidential election results here - the light blue supported the pro-Russia candidate Yuriy Boyko, who didn't advocate being part of Russia but instead wanted better relations with Russia. After the invasion, Boyko himself apparently became part of a new party of ex-pro-russian candidates who now support Ukraine against Russia.

Generally it's hard to say how many fit into the category of "wanted an independent Ukraine but that is still close to Russia", but my sense is that it was a pretty large part of the blue group. Consider for example the average American liberal who wants the US to be more like Canada - probably there are some who would rather have their part of the country become part of Canada, but it's probably not most and certainly nowhere near all. Additionally, it's likely that many who actually did want to live in Russia changed their mind after the atrocities of the invasion.

There were parts where people were more likely to speak Russian than Ukrainian, though it did not follow that they would rather be ruled from Moscow (as Russian propagandists claimed). In any case, one effect of Russia attempting to equate the Russian language with the Russian state has been a lot of Ukrainians who always spoke Russian learning Ukrainian and switching to using that in public, just to leave no doubt.

No, why would they? Haven't you heard that Russia is literally Mordor filled with orcs, while Ukraine (especially under Zelensky's rule) is a bastion of democracy? I don't think that western governments/media and especially Zelensky (who doesn't benefit at all from this war and is the main victim here) would lie to anyone about that!

Now that we got sarcasm out of the way..

If you want to form an informed opinion about it you can start by looking at the map of 2010 elections, where there was basically two candidates, a pro-EU one and pro-Russian one. Take a look at the distribution and percentage of people voting for a particular candidate in particular regions. Then look into the coup that happened in 2014, which powers participated in it (you can listen to their leaked conversations as well), then remember the electoral map you just looked at, and maybe make some conclusions.

That's "before the invasion" the part.

"After the invasion" part is of course less clear, but again if you want to start making an informed opinion you can start by looking, for example, at the level of freedom people located on Ukraine controlled territories have and that of people on annexed by Russia territories (or just "Russia proper" territories). Who is free to leave at any moment and go anywhere, and who is forbidden to leave the country because the government needs cannon fodder? Who is encouraged to volunteer to join the army with monetary incentives, and who is kidnapped from the street, beaten, held in basements for days and forcefully sent to the meat grinder? Again, you can try to make some conclusions from that, and maybe speculate whether it's possible that people from one of those groups may want to join the other one.

As a finishing touch - I am Ukrainian, I was born there, and lived most of my life there. Thankfully, I left the country before the war started. If I visit Ukraine - I will most likely be dead in a matter of months (after being kidnapped and beaten by the brave TCC guys). If I visit Russia - I will at most face more scrutiny at the border due to the war. I'm not from an eastern region of Ukraine and I never wanted it to become a part of Russia. You can try to guess which options are preferrable for me currently.

When modern-day Ukraine was formed in 1990, it was majority Ukrainian, but with a sizable Russian (and smaller Romanian and Polish) minority. Over the next twenty or so years, this minority voted for parties and politicians that favoured stronger ties with Russia. In contrast, ethnic Ukrainians supported joining (or at least aligning with) the EU. This conflict came to a head in 2014, when the pro-Russian government was overthrown by pro-EU protestors. Relations between the two groups have worsened since then, leading to pro-Russian militants seizing power in the (Russian majority) Donbass and Crimea, and joining Russia.

As much as they are still Nazis in the USA, yes.