We don't have many accounts of how the GRU and Stasi responded when they were informed of the tunnel. It's probably a safe assumption that they were unhappy to have been left so exposed, but there would also likely have been a grudging understanding of the KGB's rationale: when you have an agent as well-placed as George Blake, you'll do almost anything to protect them and keep them producing intelligence.

One thing to note at the outset: you should be under no illusion that the KGB and GRU did much cooperating or sharing of information if they could possibly avoid it. The two services were arch-rivals; the KGB was arguably the more powerful and prestigious organisation within the Soviet state, but the GRU was a large and capable intelligence service in its own right — and was largely shielded from KGB interference by the fact that it reported directly to the General Staff of the Armed Forces and the Minister of Defence.

The KGB particularly resented the GRU's extensive foreign operations; their view was that their First Chief Directorate should have the monopoly on overseas intelligence collection. In 1947, there was short-lived attempt to merge the foreign departments of KGB (then called MGB) and GRU into a new 'Committee of Information' (KI), inspired by the United States' creation of the CIA that same year. The two organisations proved impossible to integrate, and KI was disbanded within three years.

Similarly, the KGB was generally unconcerned with the feelings of its allied intelligence services, like the Stasi. Their intelligence partners in other Warsaw Pact countries were always treated as subordinates, never peers — although the Stasi was first among equals, as the most important and capable of their 'fraternal' services. Information-sharing was highly selective, and frequently monodirectional: the KGB expected full disclosure and willing cooperation from its allied agencies, but would only return the favour when it was in their interests to do so.

This, from John Koehler's history of the Stasi, illustrates how deeply-rooted KGB influence was in the Stasi:

Until the end in 1990, the KGB maintained liaison officers in all eight chief directorates of the Stasi. Each officer, in most cases a full colonel, had his own office within the Berlin MfS compound. The KGB placed special importance on Markus Wolf’s foreign espionage directorate where it occupied three offices. In addition, the KGB was ensconced in every one of the 15 Stasi district headquarters around East Germany, occupying 30 separate offices. KGB officers had access to all information collected by the Stasi. In its structure, East Germany’s Ministry of State Security had become a clone of the Soviet Committee for State Security.

So, to the question of "trusting" the KGB, or choosing whether or not to share intelligence: frankly, the KGB's lack of transparency in the case of Operation Gold likely didn't change much one way or the other. The GRU and military establishment would continue to resent the KGB, and the Stasi would continue to serve as a junior partner.

To turn to the specific questions around Operation Gold: an illustration of just how seriously the KGB took the matter of protecting Blake's identity can be found in Battleground Berlin, an excellent history of the intelligence war in Berlin co-written by the former CIA station chief, David Murphy, and the former KGB rezident, Sergei Kondrashev — the latter of whom was Blake's handler while serving in London at the time of the tunnel's conception.

According to Kondrashev, "only three persons in the First Chief Directorate were aware that a source existed": Kondrashev, the chief of foreign intelligence Aleksandr Panyushkin, and the "directorate deputy chief for operational-technical matters" Arseny Tishkov. The head of the Eighth Chief Directorate, which oversaw cryptography and communications security, was briefed on Operation Gold but was not told Blake's name.

Soviet military authorities were briefed that the US and UK were targeting military communications: in September 1954, shortly after construction on the tunnel began (but still a year after the KGB learned of the plan), the KGB sent defence minister Nikolai Bulganin a memo on Operation Silver, a similar operation that SIS had run against Soviet military communications in Vienna, and apparently warned them that Soviet forces in Germany might be targeted next. Per Murphy and Kondrashev:

The Soviet military leaders were surprised by the quality and extent of information that the Western Allies obtained on the Soviet occupation forces in Austria and Hungary by monitoring conversations of Soviet officers and enlisted personnel. Nevertheless, they took no action against the intercept operation planned for the Berlin area lest the source be endangered. Another factor in the decision not to disclose the tunnel plans outside of a narrow circle was the fact that the intercept activity had not yet begun, and Blake could continue to keep the Soviets advised of any progress. As a result, no one in Germany was briefed, not even Yevgeny Pitovranov, head of the KGB apparat in Karlshorst, until after Blake's connection with the technical section in London had ended in spring 1955.

Military leadership evidently made some oblique attempts to encourage better communications security. This is from Jonathan Haslam's Near and Distant Neighbours:

Instead, ever conscious of the damage being inflicted but mindful of the need not to jeopardise their most promising agent in place, the Soviet military authorities were allowed only to issue regular warnings to officers that they restrict the length and content of their telephone calls. Sometimes officers even received verbatim printouts of their calls as a stern reminder to remain vigilant—not that it had much noticeable effect.

The KGB's ostensible calculation that their own communications were safe (because they used separate infrastructure), and that the year-long compromise of Soviet military and GRU communications was an acceptable tradeoff for protecting Blake, proved somewhat off-base. The tunnel captured the communications of the Third Chief Directorate (military counterintelligence) detachment at Karlshorst; the First Chief Directorate rezidentura in Berlin necessarily liaised with their counterparts in GRU, the military and the Stasi, meaning they too were often speaking on compromised lines. The overall operational risk to the KGB was significantly less drastic than to the GRU, whose organisation and operations in Germany the CIA and MI6 were able to map in more detail than ever before, but the tunnel did nonetheless offer a small window into the KGB's German operations.

To put an even finer point on this irony: intercepts from the Berlin tunnel played some small role in Blake's eventual exposure, alerting British intelligence to the existence of a Soviet agent inside SIS' Berlin station. Blake was eventually conclusively identified as that agent in 1961, when a Polish intelligence officer named Michał Goleniewski defected to the United States.