You should be fine without using a tripod for normal use. If you plan on long shutter/exposure times, or plan to stitch photos, then yes, you should use a tripod. I shoot free hand all the time with my 18-55 with no issues. If you are concerned about camera movement you can always carry a mono pod. Most places don't have a issue with mono pods, but may with a tripod.
Depends on the light conditions but for general use you should NOT need a tripod. The 1.8 is a fast lens which will enable you to achieve fast enough shutter speeds for hand held shooting under normal conditions. Try and keep the shutter speed above 70th (twice the focal length) and learn to hold you camera properly and you will be ok. Night shots or long exposure shots then yes, you will need to support your camera
