Shortly after World War II the Americans did observe and censor the newly made movies in Japan. Did Kurosawa make that many Shakespeare-adaptations in that time, because it was easier to get them accepted? Or was he simply a Shakespeare-fan?
One critic's opinion is that Kurosawa was indeed influenced by censorship, both Japanese and American, during World War II, but that was not the only reason that he produced a lot of foreign adaptations:
Perhaps the most significant factor in his stylistic development as a director was WWII. At the time that Kurosawa was developing his own distinctive style the censors had great power. During the war it was the Japanese who dictated content, and after it the Occupying American's [sic] had censorial control. However, his love of foreign literature and film predated this era.
It looks like the opposite of your question might be true. Kurosawa attempted to release a film in 1943 based on a judo novel by a Japanese author, and Japanese censors deemed the film too Western in style or content. From Wikipedia:
Shooting of Sanshiro Sugata began on location in Yokohama in December 1942. Production proceeded smoothly, but getting the completed film past the censors was an entirely different matter. The censorship considered the work too "British-American" (an accusation tantamount, at that time, to a charge of treason), and it was only through the intervention of director Yasujiro Ozu, who championed the film, that Sanshiro Sugata was finally accepted for release on March 25, 1943. (Kurosawa had just turned 33.) The movie became both a critical and commercial success. Nevertheless, the censorship office would later decide to cut out some 18 minutes of footage, much of which is now considered lost.
Kurosawa began his career during the Showa war and during that time he, like all other Japanese filmmakers, was confined to making "policy pictures" that supported the militarist agenda. This necessitated a large amount of censorship. But there was no Shakespeare there and nor were there any films that would later be hailed as "masterpieces" or the likes.
After the surrender, Japanese film was censored under SCAP (1945-1952). Criticism of the American occupiers was forbidden and films like Drunken Angel and Stray Dog walked a thin line in their harsh portrayals of life under occupation. Indeed, it's likely that the fetid pond in Drunken Angel--ever present, overpowering and destroying the physical health of the neighborhood's inhabitants--is meant to be understood as criticism of the occupation and its deleterious effects on the Japanese soul.
During this time, Kurosawa made the following films:
Sanshiro Sugata Part II (1945) The Men Who Tread on the Tiger's Tail (1945) No Regrets for Our Youth (1946) One Wonderful Sunday (1947) Drunken Angel (1948) The Quiet Duel (1949) Stray Dog (1949) Scandal (1950) Rashomon (1950) The Idiot (1951)
This was an extremely fruitful time and the relationships (ie, Mifune) and themes that he established during SCAP proved the basis for the rest of his career. But no Shakespeare.
His two proper Shakespeare adaptations, Throne of Blood (1957) (Macbeth) and Ran (1985) (King Lear) as well as The Bad Sleep Well (1960) (inspired by Hamlet) were made long after the end of that period, when Japan had an independent democratic government with no censorship to speak of.
I would not call two adaptions of Shakespeare (Throne of Blood and Ran) many. And he made Ran partly because "Hidetora is me".
As a child Kurosawa read a lot. He was a particular fan of Russian literature and Shakespeare. while his early films were not adaptations of Shakespeare there was definitely a lot of influence there, for example, the manner in which he constructed his characters.
In Kurosawa there has always been a strong western influence, matter of fact one of his favorite directors and one of his biggest influences was john ford the american director.