My core question is: Why are right-wing movements in Europe currently expanding their popular support base?
From my perspective, it is essentially a reaction to political changes in Europe.
Nowadays, everything that slightly disagrees with some ideas of what left "should be" is labelled "far right" without any form of debate. Typically, being skeptical on the benefit from immigration, equity, minorities or the social context/culture explaining everyting is morally condemned and sometimes leads to socio-profesionnal ostracization.
Additionally, Europe has assumed for decades that multiculturalism is morally good and desirable, and that the other countries in the world love us and will follow our model. This led to more left-winged policies and the overton window expanding more left, giving the feeling that more things are "right-wing" while not necessarily true. Taking back the example of immigration, communist themselves were fairly against too much of it a few decades ago, whereas it only belongs to the right ideologies nowadays.
To keep it relatively short, what I mentionned affected many aspect of our life: First, most cultural policies are left and liberal, same for immigration policies and EU policies; hence the moral compass is left oriented and it has been for 30+ years.
Second, with the overton shift, I believe most mainstream "right wings" in Europe are left-wing elsewhere and we basically have uniformized, flavourless political gradient.
Third, the policies those main parties led to people getting poorer, struggling more economically, socially and psychologically. We went through deindustrialization and environmental norms that, while (sometimes) motivated by noble goals,were often applied without considering potential advert effects or without implementing compensatory measures. This resulted, among others, in more unemployement, higher energy costs (not mentioning the fiasco with Russia), and increased pressure on our welfare state and job market, especially considering immigration and EU free movement.
Saying anything about that can be taboo, hence setting the perfect ground for more political distrust and more radicalization because when you can't express yourself, it probably becomes much more tempting to assume "it's because I'm right and the problem is even bigger". Now, the expansion of (far?) right wings in Europe is, in my opinion, mostly a populist highjacking of a legitimate protestation by opportunistic and power hungry leaders.
External Influence: To what extent do distant figures like Donald Trump and Elon Musk play a role in this trend?
I suppose they only give some inspiration here, especially to make the poltiical scene more clownish than it already was? EU countries are mainly US vassals, and will comply to whatever US does, so having Trump in the US might also shape a little what EU states do. However, I believe Trump is nothing more than the product of USA too, just like the rise of the right in Europe in general.
The Role of Immigration: Are immigrants causing genuine, personal problems for ordinary citizens, or is this issue being magnified by media influence, as it often appears from afar
Immigrant are a net cost in France at least. We globally have more problems nowadays than before and it's often including migrants. For instance, migrants from some countries are overrepresented in sexual assaults, violent crimes... We also have problems with terrorism or churches burning/priests being stabbed.
Medias can sometimes influence it, but as it has been going for a few years, I would say you have a few medias talking non-stop about it (and I guess they have content for it...),, but you also have a few medias that never mention or always minimize what happened. For this last, I think very much of UK medias and institutions that covered grooming gangs for more than 40 years and, when they couldn't hide it anymore, labeled the criminals as "Asians" knowing perfectly that "Asians" would mainly appear as East Asians in our minds.
Future Outlook: How do you see the future? Do you anticipate the dismantling of international treaties, the abandonment of global trade and free markets, or potentially the start of a new phase of colonialism?
I suppose you refer to how US is moving these days? I expect nothing good. EU is cooked because our leaders are incompetents and sold us to USA instead of taking their responsibilities.
The US and Trump are desperate to show that they still are the best in the world. However, that's a last struggle and they already are admitting that their hegemony is about to end here. They are sacrificing EU to get some time (brain drain, industrial robbery, mass surveillance...) and attacking small countries like Venezuela or, worse, Niger.
For free market, it's time it ends honestly, because of how it's used, but I don't believe it will die as soon as the US. Concerning new phase of colonialism, I'd say no. Colonialism never stopped, but US is just going back to Africa a ltitle bit and more violently just to pretend. However, European powers have long been replaced by US, Russia and more recently China in Africa (China debt trapping everyone). Latin America has never escaped US colonialism either if you look at it...
We are just witnessing world being multipolar again, with economic center moving (back) to Asia and particularly China.