Doubt on Regular and unambiguous

Question

Please explain with examples,

1)is every regular grammar is unambiguous?

2) An unambiguous grammar will always generate DCFL?

Comments

For 1). Take $s\rightarrow a$ which is regular grammar and generating regular language.

Now, Is this grammar ambiguous or unambiguous .

Like this, you can try for 2nd .

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1) S -> S / a ,

  This is regular language with RE = a* , this is ambiguous language , correct?

Hence Every regular grammar is unambiguous is FALSE.

2) WW^r    , for this if we can write unambiguous grammar, since we can only parse using NPDA,hence 2 statement is also FALSE.

Please correct me if i am wrong.

Answer 1

i)

$S \rightarrow wS$

$S \rightarrow w/\epsilon$

The language is regular but ambiguous.

ii) Certain but not definite.