1. How can i spot a genuine taxi, and not the rip off ones which I need to get into the centre
The name of a company on top. Lodz airport isn't far from the centre anyway.
2. Is it safe to use the trams at night?
Yes
3. Is it safe to walk alone at night?
Yes, though the usual precautions are sensible.
4. Would their be any recommendations to visit towns outside of Lodz(for sightseeing), but within close proximity to Lodz itself
No. There isn't much worth seeing in the surrounding area, but a trip to Warsaw might be a good idea.
5. Where am I able to buy a sim card upon arrival at the airport
At any newsagent or kiosk. They are labelled either 'Ruch', 'Kolporter' or (the best ones) 'Relay'. There is one at the airport, and the airport one is most likely to have an English speaker in.