I put the zinc metal into sodium carbonate solution.
As there is water reagents in it, will it be:

1. Zn + H2o --> ZnO + H2 or
2. Zn + 2H2o --> Zn(OH)2 +H2

It is the second reaction that will occur.

Water dissociates as follows:

H_2O \rightleftharpoons OH^- + H^+

You cannot have Zinc oxide in theory.

Then it will be more applicable to say
Zn + 2H2o --> Zn(OH)2 +H2

Only hydroxides will be formed?

I would say yes. But since in practice, other things can happen, I cannot say for sure that no ZnO at all will be formed. Tests like mass spectrography may be used to confirm the presence of ZnO, if ever it was present.

If that's the case, then does the hyrdoxide be present in the solution or on the metal?

I can't say either, unless I had done it in front of me. I did see zinc and acid reacting, but not water, since it is quite a slow process. I guess that it goes into solution though. Try doing it in your lab.

then it will be more applicable to say
Zn + 2H2o --> Zn(OH)2 +H2

only hydroxides will be formed?

If the water is acidic, a salt of whatever acid is present will form. Water is often slightly acidic (carbon dioxide), so you may get some zine carbonate.

Unknown008 is quite correct that you won't get the oxide -- at least not from a solution. The oxide will form if you dry out the hydroxide (precipitate) in a hot enough oven.

Zinc doesn't react very rapidly with pure water. That's why it is used in galvanization. I think that it is similar to aluminum and magnesium in that it's reactive enough to form a hard zinc hydroxide coating on its surface that protects the metal.

ın experiment zinc does not react with water but therotically the zınc hydroxide will be formed.