well this one is pretty messed up information.
first after you defeat misty, you should talk to billy on route 25 to get a ss anne ticket later in vermilion,
then after you get the ticket, cerulean guard on east side is gone and you can go south to vermilion, no need cut.
inside vermilion, you need to go to ss anne and finish it's quest to get HM Cut so you can fight 3th gym,
after win the gym, the path to east open, where digglet cave first entrance connected to route 2 for HM Flash.
and about dig, you need 150 happiness to the pokemon so you can use it, plus if you havent interact with both side of dotted hole, you cant use the dig to pass the cave (so you have to go pass the cave at least once to reach 2nd dotted hole)
for future reference please use this link
tinyurl.com/prowalkthrough