If my landlord has stopped paying utilities, which are included in my rent, if I pay the bill can I legally deduct the cost from my rent?

You can make such a deduction, however, if the landlord is so financially strapped as to now be unable to pay utilities, you just may want to move.  What if a repair is needed?  And it's a good bet that normal maintenance of the property is not being done.

You can lawfully terminate your lease on the basis of "constructive eviction".  Without utilities you cannot continue to live in the premises.  This also constitutes a breach of the "warranty of habitability".

Just be sure to give your landlord written notice of your intention to vacate and your reasons why.  Send the letter to them, certified mail, return receipt requested, and keep a copy for yourself.  You should be fine but if they try to take you to court it will be good evidence of your current situation.

You can do either one.  Under California law, a landlord must only rent an apartment or dwelling that is habitable.  In today's age, habitability includes electricity.  Therefore, you can either consider it a constructive eviction or material breach of the lease and leave.

You have another option and that is to pay it yourself (pre-supposing the electric company will allow this) and stay in the unit and deduct the payment from your rent.  If the payment is greater than your rent, than keep simply paying the utility and keep deducting from your rent.  However, do not put the utility in your name.  It could pose many problems for you in terms of the contractual agreement overall.