Does Islam teach establishment of shari‘ah in society?

An interesting historical parallel is that there was an aversion to shari‘ah at the spread of Christianity; similarly, there is an aversion to shari‘ah now at the spread of Ahmadiyyat as well. History sometimes repeats itself. Hadrat Musleh Mau‘ud ^ra explained:

When Christianity spread in Rome, people were not willing to relinquish their national constitution. They considered Roman law superior to every other law, in fact, European governments still benefit from Roman law. On the one hand, they were attracted by the alluring teachings of Christianity, but on the other hand, their superiority complex concerning Roman law prevented them from bowing before Judaic law. Nevertheless, the Romans found an excuse, and they misinterpreted these verses to mean that religion should be followed only in matters of prayer and fasting, but not in worldly matters, for religion has no authority over these matters. It is up to people to draft the law of their choice.

The present interpretation of “separation of church and state” in the West is originally based on misguided Christian ideology. Hadrat Musleh Mau‘ud ^ra explained:

In short, they detest such commands of religion as are related to the organization, and thus, they declare the shari‘ah to be a curse… To them, religion has no control over affairs of the world; rather, a country can itself come up with rules and regulations of its own choice 

(Khilafat-e-Rashidah, p. 14–18).

However, this is not the Islamic interpretation of “separation of mosque and state.” Misguidance found its way into Christians because they did not have Khilafat. However, the difference in Ahmadiyyat is that we have Khilafat to prevent such misguidance from entering Islam.
Hadrat Musleh Mau‘ud ^ra was once asked, “Concerning the punishments for crimes in the shari‘ah that are set in the Holy Quran, can the Imam of the time, in some circumstances, abolish them and appoint other punishments in their place?” He ^ra answered:

There is no such law that needs to be abrogated and no commands of any new shari‘ah need to be brought in its place. This is the perspective of those people who are afraid to speak against Western thought. These people are opportunists, whoever has greater worldly influence, they bow their heads before them. Ask such people and write to me as to which commands need to be changed and cannot be acted on at this time.
However, according to Quranic principle, ignorance does lighten or abolish a punishment. Thus, if a people have recently entered Islam or are in an age of ignorance, then until their education is complete, the Islamic ruler can be lenient in punishments. And this is not changing the shari‘ah; rather, it is a stage in the progress of education 

(Fatawa Hadrat Musleh Mau‘ud, vol. 2, p. 104).

Laws about governance were revealed through the Holy Quran because humanity is in need of a perfect teaching on both an individual level and on a social level. It is the responsibility of Muslims to persuade the world to accept both the spiritual and the secular teachings of Islam and to establish them in the world. Hadrat Musleh Mau‘ud ^ra said:

The Muslims, who are bound by Islamic directives in a purely religious organization, are also not free in the matter of the organization of state, and are under obligation to submit to the organization of the state established by Islamic shari‘ah. Even one glance at the Holy Quran and sayings of the Holy Prophet ^sas shows manifestly that Islam is not among the religions of the first category mentioned above but is of the second category. Islam issued commandments not only for certain beliefs and individual behavior, but also for government and law… This is because the God Who commands us to pray, fast, perform Hajj, and pay Zakat has also given guidelines for the political affairs and organization of a country. It cannot be said that every nation and every country is at liberty to invent an organization of its own choice; rather, it has to follow Islamic rules in all spheres of life.

(Khilafat-e-Rashidah, p. 20, 21–22)

The objective of the Ahmadiyya Muslim Community is to establish the shari‘ah of Islam, but with persuasion, with democracy, and without infringing on the inalienable religious freedom of any people.