Given that the star is crossing the local meridian line in a certain location, I've tried calculating the altitude of a star by finding the difference between the declination of the star and the latitude of the position from where it is observed (which I believe is not correct). I'm so confused about that and it would be of great help if someone could explain the altitude calculation. Also, I'm wondering how to calculate the position of a place (latitude and longitude) while the declination and the right ascension of a star at the zenith is known.
For the second question: If you know the coordinates of the Zenith, your latitude is exactly Zenith's declination. For your longitude you can not rely on the Zenith: the same star will be at the Zenith at the same sidereous time for every place in the same latitude, so you need a clock besides the telescope. cf. http://en.wikipedia.org/wiki/History_of_longitude#Problem_of_longitude
For the first question: The altitude of a star that is crossing the local meridian is also quite easy.
For Northern Hemisphere, a star that is crossing the North local meridian (that is, between the Zenith and the North Horizon) the latitude is $90-alt=dec-lat$
For Northern Hemisphere, a star that is crossing the South local meridian (that is, between the Zenith and the South Horizon) the latitude is $90-alt=lat-dec$
Same goes for Southern Hemisphere if you change also North Horizon and South Horizon.
You did not asked specifically for the Azimuth, but again you need a clock for that.