Look at your edit curve menu.
Inserting points is Edit Curves ->insert knot the options box gives options to insert at selection or number of points between selections. Since maya works on exiting selections select curve parameters by holding rmb above curve and selection curve point. Now theres no delete segment but here is a Edit Curves -> detatch curve command that does the same thing. Or you can copy the curve over and delete the extra points etc. etc.
Tough in reality you can skip the insert knots part alltogether and detatch the curves directly. So select 4 points and just detach, then delete the newcurves that lie in between, (no it wont alter the shape of your curve) as thats one step less to work. If you want you can go further and eliminate the delete steps too but thats another story. So not only could maya do this but it could eliminate 25% of mouse clicks by default and even eliminate more if you wished.
And all of this you should have been able to figure out yourself by looking at your menu structure. Learn to look at what you have.