If you're relatively new to cycling or are increasing your training significantly it's not unusual for your MaxHR to increase.
Most sources say that, if anything, max hr actually decreases with training. The numbers you see on the monitor are likely to increase with training, because of the confidence and ability to push your HR that high. Even in a clinical setting, where the point is to get people to VO2max (and max HR), untrained people will rarely actually reach this point. It's damned hard work.
Most people have a different MaxHr for various exercises, i.e. running Max will be different than cycling Max which is different than cross country skiing max.
Yup. cc skiing >running >cycling> swimming.
At the end of the day it's a pretty meaningless number that has limited value for training.
+1. OP, the only thing this is really needed for is for doing submax estimates of VO2 max, which really aren't very useful at all.