Clearing all of this...
If you attempt to activate a card effect whilst being unable to fulfil the effect, it is considered an illegal activation and you will have to rewind the game state until you're back at the starting point.
So, in this example, Player A cannot draw 2 more cards, even if it would cause him to win with Exodia.
About decking out, no optional effects that would cause a Player to run out of deck cards, like Pot of Greed, Hand-/Card Destruction, and so on. Mandatory Effects, however, still have to activate hence their nature of being mandatory.
Example for this would be Player A Synchro Summoning a Synchro Monster while Player A already controls T.G. Hyper Librarian with no cards in his/her deck: T.G. Hyper Librarian is a mandatory effect, so it will attempt to draw a card in all circumstances, even if Player A cannot draw anymore, causing Player A to lose, against what Placido said above.
Othet example for such a mandatory effect that would cause deck out is Morphing Jar; all FLIP Effects are mandatory effects, Morphing Jar's effect resolves in any circumstance and, if either Player A or Player B has under 5 cards in his/her deck, the respective Player will lose.
Note that here a tie is possible, if both Players have under 5 cards in their decks.
To paying costs...
Life Point costs are generally, in the TCG, not able to be paid if the amount is equal to or higher than your current Life Points, even if paying those Life Points would cause something to happen which would either have you gain Life Points (we will just assume that there is a card effect which will immediately have you gain Life Points after paying them) or would cause your opponent to lose (we will just assume that there is a card effect which will immediately cause damage to your opponent; Fire Princess cannot be this example due to it going on chain).
In the OCG, however, you are able to pay Life Point Costs which equal your Life Points (you are still not able to pay costs higher than your Life Points).
Examples would be activating and successfully resolving Solemn Warning while only having 2000 Life Points; summoning a Synchro Monster with Wicked Rebirth while having only 800 Life Points (I will mention this later on); paying 800 Life Points to summon any monster from your graveyard with Premature Burial; and so on.
However, as mentioned above, we are able to assume that doing so will still cause the current chain in which the Life Point Costs were paid to activate a card effect to resolve properly, means that if, only if, a Player activates Master Cure - Dian Keto and then chains Wicked Rebirth while only having 800 Life Points, after resolving the chain properly, that Player will then gain 1000 Life Points and does not lose the duel.
This, however, is an assumption based on the Yu-Gi-Oh! 5D's anime where indeed a Player paid for Wicked Rebirth while having only 800 Life Points left. The monster still was summoned and passed on to the next Player (assume the duel was a Turbo Duel).
I do not guarantee that what I said about the resolving chain link with Dian Keto and Wicked Rebirth to be true since I never had to deal with such situation, but I can still assume it to be as written above.
Anything else you need?