There are varying opinions on this one. But, if your dentist, with the assistance of radiographs, feels you should have them removed for preventive reasons then it is a good idea. You could probably leave them and risk infection, periodontal problems, caries or pulpal issues that may take years (or sooner) to develop or nip it in the bud and extract them. Ultimately the decision is yours.
if the gum tissue is consistently irritated despite good hygiene efforts, that's good enough reason for me to remove them. the potential to form cysts, however is not. if there are no cysts now and you go for regualr checkups that situation can be monitored, and it is not so likely to happen as to warrant extraction of those teeth "just in case" cysts form. The other thing to keep in mind is that wisdom teeth are difficult to keep clean, especailly in front of and behind them (which is why your gums are irritated) .cavities form more easily in these teeth, and on the back surfaces of the molars in front of them. if that happens, now you would have to remove the wisdom teeth anyway simply to gain access to the back surfaces of the 2nd molars in order to remove the decay, which is difficult anyway due to their location in the mouth. If the extractions do not seem to be complicated, and risk of complications is low, if it were me i'd remove them if you cannot control the irritation and keep them clean/cavity free.
If I do proceed with having them removed, do I face any problems with shifting of other teeth?
nope... teeth drift forward, not backwards.