Jewish Talents

Depiction of Maimonides, a great Jewish Rabbi and prominent doctor appointed to King Saladin, adversary of Richard-1.

The question was:

Why are there so many Jewish lawyers, doctors, politicians, and actors?

Here are some observations that may help answer some of the above questions, as to why Jews often succeed in certain professions:
===============================
Jewish Lawyers:
The Jewish Religion is involved with Religious Law. Being religious in Jewish terms primarily means learning laws, and how laws are deduced, what are the principles behind the laws, what are the pros and cons in adopting certain legal principles, what happens when different laws conflict with each other, etc. All this is similar to Secular Law and basically uses parallel frameworks of deduction, etc. In addition much of Anglo-Saxon legal tradition is either derived from Jewish Law or evolved along the same lines. It was prophesied that out of all the Israelite Tribes those of Levi, Judah, and Issachar would have the greatest aptitude for the Law.
It was prophesied that the Jews would keep the Law and be Lawmakers.

Genesis (NASB) 49:
 10 The scepter shall not depart from Judah,
 Nor the ruler's staff from between his feet,
 Until Shiloh comes,
And to him shall be the obedience of the peoples.

The word translated above as "the ruler's staff" in Hebrew is "MeChokaik"" This literally means "Maker of Laws" i.e. Lawyers, etc.
 The KJV translates it as "Lawgiver." So do some of the other translations and this is what the Hebrew original literally means.
Even when Jews become less observant they often pick up Jewish traditions and modes of thought and perhaps an inherited aptitude for legal matters.

cf.

Psalms 60:
 7 (9) Gilead is Mine, and Manasseh is Mine;
 Ephraim also is the helmet of My head;
 Judah is My scepter.

Here too the word translated as "scepter" is more correctly rendered as Lawgiver.
 The verse is repeated in Psalm 108:9.

A related point is that the future Messiah will emerge from Judah. The Jews have a Messianic Complex and often want to reform what needs reforming. This attitude may sometimes be misguided in practice but it exists.
 In the end it shall be appreciated:

Zechariah 8:
 23 Thus says the LORD of hosts, 'In those days ten men from all the nations will grasp the garment of a Jew, saying, 'Let us go with you, for we have heard that God is with you.' '

Levites in Judah as Lawmakers.
Many Jews belong to the Tribe of Levi. When faced with legal problems it was a commandment to consult the Sages most of whom (according to one tradition) were Levites. This is indicated in the Torah.

Deuteronomy 17:
 9 And you shall come unto the Priests [i.e. Cohens from the Tribe of Levi], the Levites, and unto the judge that shall be in those days, and inquire; and they shall show you the sentence of judgment:

Even today, among the Rabbis those of them who are Levis to a very disproportionate degree are the ones who make the legal rulings.

Deuteronomy 33:
 10 They [i.e. the Levites] shall teach Jacob your judgments, and Israel your law... 
 ===============================
Doctors

Jews relatively speaking are merciful. They have a Messianic Complex and want to put right what is not as it should be. They are intellectually inclined and scientifically curious. They are also human beings and not against status and a reasonable income.
 The Medical profession has the potential to supply all of this.

Several great Rabbis were also prominent doctors, e.g. Maimonides, Nachmanides.
Maimonides ascribed a religious dimension to the calling of medicine.

Quotes from Maimonides:
Give a man a fish and you feed him for a day; teach a man to fish and you feed him for a lifetime.
The risk of a wrong decision is preferable to the terror of indecision.
Teach thy tongue to say 'I do not know,' and thou shalt progress.

===============================
Politicians

The choice of politics emanates in part from characteristics dealt with above. Legal matters and politics are interwoven. If you deal with one you shall often need to be involved with the other. Politicians need some knowledge of the Law: Lawyers often see were the Law needs fixing and develop an urge to fix it. So too, the Messianic Complex, mentioned previously, leads to Politics.

===============================
Actors

In Talmudic times the Sages were vehemently opposed to the theater and acting. It was considered an agent of Hellenization and a corrupting influence.
Jewish education however could develop qualities that might be useful in the acting profession. Examples include: learning text by rote; putting oneself in the shoes of others when considering legal and moral matters; empathy for others; demonstrating through facial expressions and voice intonations different emotions; etc. Also music. David was a musician. David composed the Psalms to be set to music. David also invented new musical instruments. 
cf.

Amos 6:
 5 Who improvise to the sound of the harp,
And like David have composed songs for themselves,

The expression translated here as "songs" in Hebrew is "cli-shir" i.e. musical instruments. David invented new musical instruments!

The Levites, who became part of Judah (2-Chronicles 11:13), had their own orchestra of musical instruments and a choir of singers in the Temple Service. [Note by Ami Elat: The Judeans were so well known for their music and singing, that the description of the exile to Babylonia included a mention of singers and musicians].