How long does one normally see a comet in the nightsky for? From some news articles on astrobiology websites, it seems possible to view parts of a comet and its orbit for around a week, though most seem to use telescopes...

This depends on several things: the size of the comet; its activity (release of volatiles); it's orbit in relation to distance from the sun and to the Earth; etc. The brighter comets can be seen by the naked eye for several weeks. Telescopic observation is possible for months.