Nope some kids are born with teeth

I cut my first two teeth right before 4 months, fortunately my kids so far have not cut teeth until 8 months
Some babies find cutting teeth very painful, very disruptive, etc... while other babies have no problems whatsoever and mum and dad get surprised when the tooth pops through! But it sounds like you have the classic signs on your hands, the sleeping can probably be helped with Bonjela. Be aware that there's a growth spurt around this age too.