.... the connection won't be as secure. connectors tend to have a range of sizes that it can fit, it will say on the back. like 12-16awg or whatever.
radioshack/homedepot/lowes/frys all have plenty of options. heatshrink is nicer than tape though.
wirenuts are only good for thicker wires.
as for nicking wires. basically you make the nicked spot thinner, so effectively its higher gauge bit at that point which can lead to overload/failure point.
and you can always hotglue/ziptie for extra reinforcement to take strain off the connector.