Is the resistance of nichrome wire affected by its temperature?
The position is a little more complex. Yes, resistance does change with temperature. But the resistance of nichrome, which is an alloy containing nickel and chromium (of course) actually changes very little with temperature compared with most metals. If you do the experiment suggested above (always a good idea to check these things if you can) the current will drop but at most only by a few percent. If you were to do it with a pure metal, such as tungsten, you might expect the current to drop to a quarter (or even less) of the value it started at.