In addition, and pretty specifically I suppose, I am wondering whether 17th century Dutch playwrights such as Joost van den Vondel (who has been called the Dutch Shakespeare) and Jan Six would have been aware of Shakespeare and his body of work so soon after Shakespeare's death.
I am a Spanish to English translator, so I am quite interested in the history of Shakespeare's work translated into Spanish. There were no Spanish Shakespeare translations until over a century and a half after his death, the first of which was Hamlet in 1772; however, as with many translations of English works in Spain in the 18th and 19th centuries, it was an indirect translation from the French translation (1798 saw the first direct translation of Hamlet). Other works also followed this pattern, such as Romeo and Juliet (first translated via French in 1803 and directly from the English in 1872) and Macbeth (first translated via French in 1803 and directly from the English in 1838). King Lear, on the other hand, remained untranslated until 1870 (apparently directly from English).
Unfortunately, I don't have direct information about his critical reception in Spain during the 18th and 19th centuries, but considering how common retranslations of his work were throughout the nineteenth century, it can be reasonably concluded that he was received well in Spain, despite taking a while to arrive there.