Was it necessary for the Gentile believers to live strictly by the Jewish law?

Answer 1

1) The large majority of the Torah's commands such as ritual purity, Shabbat and keeping kosher, are obligatory only for Jews.
2) Non-Jews are expected to learn from the Torah's moral imperatives such as mutual respect (etc.), and worldwide traditions such as the abhorrence of cannibalism (etc).

Answer 2

The mitzvot (commands) of the Torah may be divided into three categories:
1) Those mitzvot that are obligatory only for Jews. This category includes the majority of the mitzvot of the Torah.
2) Those mitzvot that are obligatory upon non-Jews as well as Jews. This category includes seven commands (Talmud, Sanhedrin 56a):

a] not to eat from an animal which is still alive;

b] not to curse God;

c] not to steal;

d] to maintain courts of law;

e] not to commit adultery or incest;

f] not to worship idols; and

g] not to murder.

3) There are certain mitzvot which are not explicitly commanded for non-Jews but which non-Jews are nonetheless expected to learn from and adopt. This category includes such examples as:

a] going in the ways of God (Deuteronomy 28:9);

b] giving charity (Leviticus 25:35), and other ethical teachings (such as Leviticus 19:13-18);

c] learning from the lessons of history (Deuteronomy 32:7);

d] not to ignore reproof (Deuteronomy 10:16);

e] not to mindlessly follow mass behavior (Exodus 23:2); and more.

4) There are some things that are the universal tradition of mankind, so much so that the Torah doesn't command them at all, and considers them as being self-understood. These include burying the dead, not eating human flesh, belief in the afterlife, and more.

See also the Related Links.

Link: The example that Judaism set

Link: Jewish ethics

Link: The afterlife