Does "cards with the same name can't be played" affect tokens?
Can I "play" a token?

No. You can still produce zombie tokens even if you are not allowed to play Zombie. "Play" has a very specific meaning in Magic:


  701.11b To play a card means to play that card as a land or to cast that card as a spell, whichever is appropriate.


When instructed to name a card, can I name a token instead?

It depends. If a token has the same name as a card, then you can name it. Here are a few examples:


Pack Rat creates tokens that are also named Pack Rat.
Meloku the Clouded Mirror creates Illusion tokens, but Illusion is also the name of a card.


Zombie is not currently the name of a card, and so Zombie is not a legal choice when instructed to name a card. However, if Zombie were printed as a card in the future, then it would be a legal choice.


  201.3. If an effect instructs a player to name a card, the player must choose the name of a card that exists in the Oracle card reference (see rule 108.1) and is legal in the format of the game the player is playing. (See rule 100.6.) If the player wants to name a split card, the player must choose the name of one of its halves, but not both. (See rule 708.) If the player wants to name a flip card’s alternative name, the player may do so. (See rule 709.) If the player wants to name the back face of a double-faced card, the player may do so. (See rule 711.) A player may not choose the name of a token unless it’s also the name of a card.