In this case, it's basically like in the EP.

What I was told is that U.S. Congress staff don't work for one party or the other but for both, but that might have been wrong.

One way would be to follow the relevent EU news media, such as Politico  and Bruegel (for example their podcasts). You can also watch all committee meetings on the EP's website, find all the agendas and information about the EP's files on the respective committee's websites and here: https://www.europarl.europa.eu/legislative-train/, although this is of course a very time consuming way of learning about it.

Regarding  the missing right of initiative, I am not sure whether it is often the case that the Commission simply doesn't address an issue that a large part of the Parliament thinks is important. And even if this situation came true, the degree of unity in Parliament would probably not be sufficient to coordinate the kind of blackmail you talk about. Some parties could always "defect" and blame the other parties for blocking a good proposal. And although there generally is very little media coverage about most EU legislative projects, parties are very sensitive to the possibility of bad press and don't want to be perceived as the ones blocking policies. But there can be interinstitutional compromises (for example there are formal and legally binding interinstitutional agreements between the three institutions).

Regarding the relationship between Parliament and Council: It's quite bad. There is less conflict between the important groups in Parliaments than between the Parliament and Council (or between Parliament and Commission). Parliament as a whole is more pro-european  and progressive than the Council while the Member States' positions (represented by the Council) are more driven by Member States' domectic interests, concerns and political incentives. (Often, Parliament also feels that the Commission is more on the Council's side than on Parliament's side.) However if both Parliament and Council agree that the Commission should propose a regulation or directive, I don't think the Commission is likely to resist the combined pressure. There is a provision that seems like it could give a lot of power to the Commission:  At the trialogue stage, when Parliament and Council negotiate on a particular file with the Commission's intermediation, the Commission can withdraw the legislative proposal. In practice, I don't know whether the Commission makes use of this power. 

Regarding citizens' initiatives: I am not sure about the details, but I think citizens' initiatives are addressed to the Commission only and compel it (in case of sufficient signatures) to propose a relevant regulation. Parliament can't adopt citizens' initiatves.

Really? From what I have heard (from an advisor in the EP), the U.S. Congress doesn't have group advisors. Instead, the research service plays a more important role. I was told that the Secretary Generale of the European Parliament wants the EP to be the same, which is  why the EP research service was founded.

It's certainly not a requirement. I was surprised by how many interns of MEPs did not belong to their MEP's party (maybe 50% or more ). (There was even one case of a Conservative MEP who had a Green intern.) This is even more true for the interns who work for the group as a whole rather than an individual MEP. However, of course, some MEP interns had  got their internship through party connections with that MEP. Party membership certainly raises the odds significantly even without knowing MEPs.