Base Feint Action benefits
To understand this, you have to take a look at how Feint normally works:
If successful, the next melee attack you make against the target does not allow him to use his Dexterity bonus to AC (if any). This attack must be made on or before your next turn.
In other words, when you perform a Feint, until the end of your next turn, the first attack you make against the target ignores his Dexterity bonus to AC.
Greater Feint additional benefits
Note the wording from Greater Feint (emphasis mine):
in addition to losing his Dexterity bonus against your next attack.
This part of the Greater Feint refers to the normal effects of Feint (mentioned above in this answer). Which simply means that you still get the effect of the basic Feint action when you also have Greater Feint.
So, in addition to the normal effects of Feint, the extra benefit that Greater Feint provides is that the opponent is also denied its Dexterity bonus from attacks made by anyone else until the start of your next turn, as clarified in the FAQ
Greater Feint makes the target lose its Dexterity bonus against all melee attacks by anyone until the start of your next turn, not just you.
There is a significant difference to note here. The basic benefits of Feint last until the end of your next turn while the extra benefits of Greater Feint last until the start of your next turn.
Greater Feint and Base Feint together
These can therefore be seen as 2 separate effects you now have when you feint during a turn:
- The opponent is denied his DEX bonus to AC against attacks done by anyone. This lasts until the start of your next turn.
- The next time you attack the opponent, that target is denied his DEX bonus do AC. This lasts until the end of your next turn.
Your specific questions
To answer your specific questions:
If using Greater Feint, you can still attack during your next turn and get the benefit for your first attack as usual.
If you make multiple attacks during your next turn, only the first attack will benefit from ignoring the DEX bonus to AC (and apply things like sneak attack). This is because the normal effect from feint still stands, but not the benefit from Greater Feint.
Further clarification and examples
For further clarification, consider the following cases:
- If you feint, then make an attack before the start of your next turn (for example an opportunity attack), you will have spent the basic benefit of Feint, and during your next turn you will no longer be able to make any attack ignoring the opponent's DEX bonus to AC.
- If you feint, then make several attacks before the start of your next turn, all of them will benefit from ignoring the DEX bonus to AC due to the effect of Greater Feint. However, like the previous the case, during your turn you will not have this benefit anymore.