It is not necessary to be in the state of wudu to drink zam zam.
An extra point of benefit : Excellence of remaining in state of wudu at all time and praying 2 units of prayer (in non prohibited times) after completion of wudu with reference to the practice of Bilal r.a. Called tahiyyat-ul-wudu.
Is it permissible to eat in the tableware in which Alcohol or other haraam food was served previously?
In short, yes, as long as the tableware has been washed properly.
For cleaning utensils, in the Hanafi Fiqh, it is not necessary to wash three times or more as such. It can be washed once under running water (tap) with soap or in a dish washer to get rid of its impurities. These days all the restaurants are required by the law to wash their utensils cleanly and in the process the utensils gets clean thoroughly. Thus the utensil will be considered cleansed from haraam food remains / alcohol.
In the past, utensils were mostly earthenware which can absorb some of its content. Hence the necessity to wash it multiple times. However, the utensils nowadays are impervious and cleaning it is relatively easy.