Some cell service will work, it depends if the local tower is operational or not.

Digital phone service still requires a line with working equipment.

Most internet equipment is on UPS's, then some with generators. The ups holds the equipment untill the gen starts up if there is one, but then they usually can't support all the equipment and the heating and cooling systems required for them and they don't have a limitless power source. Big companies with the $$$ will have their equipment in seperate locations in hopes that both don't go down at once.

Yep, be lucky you were as prepared as you were so you could work around any problems that you weren't prepared for.