Hi there. Although I can't speak for the entirety of Europe and you seem to have very many questions, I can nonetheless point you towards a number of answers of mine which might be an interesting starting point for you.

In this discussion about the Italian Communist Party's relationship with the USSR, I delve into the topic of the Red Brigades in this answer on that same thread.

I also wrote this answer about Eurocommunism which you might be interested in.

I can say I have never really read anything to suggest there was much coordination between terrorist groups in Europe in the 1970s, and certainly mainstream parties had little or nothing to do with them. What is certainly true is that in the 1970s, economic instability and mistrust of the establishment led to radicalization and subsequent reemergence of armed extremism on both left and right on a scale that had not been seen since the 1930s and 1940s.

However, it's more correct to say that extremist violence, in Italy at least, brought the left and right together: with Italian Communist Party (Partito Communista Italiano, or PCI) leader Enrico Berlinguer floating the idea of a “Historic Compromise” between the Communists and the ruling Christian Democracy Party (Democrazia Cristiana, or DC) in order to set up a coalition government with Communist participation. In order to create space for compromise, Berlinguer adopted a new and lofty platform with unclear economic goals and vague commitments to international cooperation on issues like the environment, eradication of disease, and combatting hunger. Berlinguer also definitively broke with the Soviet Bloc, declaring in a high-profile 1976 interview in the Corriere della Sera newspaper that his “Eurocommunism” was unconditioned by the regimes behind the Iron Curtain, and that at any rate he felt safer living under the umbrella of NATO than he would be under the Warsaw Pact.

Berlinguer's overtures found a strong consensus among the left-leaning currents within the Christian Democracy Party, and former Prime Minister Aldo Moro positioned himself as a major sponsor from within the Christian Democrats' own ranks. Following the 1976 election, the PCI’s new platform was rewarded with 34.37% of votes, their best showing ever. The DC, in spite of increasing criticism by the press and prominent business leaders during the election, retained their share of voters at 38%.

DC kingpin Giulio Andreotti — prodded by former Prime Minister Aldo Moro — was obliged to assemble a “Government of National Solidarity” with a multitude of technocrats holding prominent ministerial positions, and the PCI agreed to abstain from votes of confidence.

The "Historic Compromise" would remain in place with Andreotti at the helm until 1979. But as the Communists moved to the center, violent extremist groups became bolder: 1976 saw the increase of a wave of terrorist activity. As you problably well know, kidnappings and shootings targeted politicians, judges and prosecutors in addition to businessmen and police. Most dramatically, in March of 1978 the Red Brigades kidnapped and subsequently murdered Aldo Moro, a figurative and literal bullet to the heart of the "Historic Compromise."

Moro’s kidnapping and murder was a catastrophic failure of Italian institutions on all levels and shocked the country. But in the 1979 elections, voters would choose to punish above all others the PCI. The “Historic Compromise” had promised economic stimulus and social stability, but voters had seen the opposite: fiscal austerity and an increase in extremist terrorism. The DC emerged from the 1979 elections stable at 38%, but the PCI fell to 30%.

In spite of the electoral setback, the breakdown of the historic compromise only invigorated the Italian Communist Party's hopes of forming a coalition government. The PCI almost managed to hammer out an agreement with the Socialist Party (Partito Socialista Italiano, or PSI) which had registered a stable 9.6% of votes in the last three elections; however at the last minute the Social Democrats (PSDI) who would offer the last parliamentarians needed for a majority defected to the Christian Democrats, who could set up a government with former Minister of the Interior Francesco Cossiga at the helm. But Cossiga maintained the controversially hard line against radical terrorism he had followed as Minister of the Interior, and the strategy continued to be ineffective: 1980 saw 125 lives claimed by radical terrorism, the blackest year yet in Italy’s fight against extremism. In fact, 1980 saw a pipe bomb set off by right-wing terrorists at the Bologna Railway Station cause the deadliest terrorist attack to ever have taken place in Italy.

Cossiga's premiership would last little over a year, as mounting political violence and an unconvincing stimulus package earned the Prime Minister a vote of no-confidence in October of 1980, prompting a reshuffle. Arnaldo Forlani, the former Defense Minister and Foreign Minister under Andreotti, was placed at the head of the government. True to the trend that the "Years of Lead" actually brought right and left together, Forlani's government also included the Socialists, and consequentially enjoyed a wider majority than Cossiga.

Although Forlani's government was brought down by a ground-shaking banking scandal after only eight months, the following years would see much more political turnover as the DC relinquished the Prime Minister's office; first in favor of Giovanni Spadolini, the head of the Republican Party (a small liberalist party) and eventually to Bettino Craxi, the larger-than-life head of the Sociality Party, signaling a true opening to the left (albeit with regards to the increasingly centrists socialists, and not the Communists). The rapid petering out of extremist violence in the early 1980s can be attributed to a combination of domestic economic reforms as well as changes in the international landscape (including economic growth born out of the accelerating european integration process) but most importantly, a willingness by large and small parties on both the left and right to come together.

If you'd like to read more about modern Italian history, do pick up:

Paul Ginsborg's A History of Contemporary Italy, 1943-1988, and Mark Gilbert's European Integration: A Concise History for a more european perspective.