Is Wittgenstein's claim to have "solved" all important questions generally accepted in the world of philosophy? If so, why do people continue to practice philosophy? If not, why not?
Wittgenstein's work is conventionally divided into early and later phase. The representative texts of these two phases are Tractatus (early work) and Philosophical Investigations (later work). Since your question about philosophy refers to conception of philosophy in Tractatus, I'll omit his later conception of philosophy.
It is a common misconception that Wittgenstein with Tractatus ended practice of philosophy. That simply doesn't hold. Let me explain. In Tractatus one must distinguish between two conceptions of philosophy. First conception represents both, traditional philosophy and philosophy in Tractatus. This conception of philosophy is traditionally called metaphysics. The problem with metaphysics is that it tries to say what can't be said but only shown. This results in plain senseless propositions of traditional philosophy and also of Tractatus. Consider what Wittgenstein wrote just before the last thesis:
6.54 My propositions are elucidatory in this way: he who understands me finally recognizes them as senseless, when he has climbed out through them, on them, over them. (He must so to speak throw away the ladder, after he has climbed up on it.) He must surmount these propositions; then he sees the world rightly.
Compare this to his remark in the preface:
On the other hand the truth of the thoughts communicated here seems to me unassailable and definitive. I am, therefore, of the opinion that the problems have in essentials been finally solved.
Wittgenstein thought that he solved questions and problems that belong to this conception of philosophy -that he finally ended with "metaphysical swamps".
But what about the second conception of philosophy? What is it? It answers your second question. The primary task of every future philosophy is now elucidatory. Philosophy becomes a skill, practice. Philosopher doesn't offer any new theories; he reveals nonsense:
4.112 The object of philosophy is the logical clarification of thoughts.
Philosophy is not a theory but an activity.
A philosophical work consists essentially of elucidations.
The result of philosophy is not a number of “philosophical
propositions”, but to make propositions clear.
Philosophy should make clear and delimit sharply the thoughts which
otherwise are, as it were, opaque and blurred.
This is the explanation why Wittgenstein ended philosophy (solve all important questions) and how is it possible that philosophy still matters - even more than before. It is also the fundamental belief for his later conception of philosophy.