Are these correct?

Bir fare gibi bir sıçan
A mouse is like a rat

Bir maymun gibi bir köpek değil
A monkey is not like a dog

(Bir) Fare (bir) sıçan gibidir.

(Bir) Maymun (bir) köpek gibi değildir.

I think that you can delete bir from sentence.

My try...

 In this type of sentences, instead of the particle "gibi", we often prefer the verb "benze(mek)"

1) Fare sıçana benzer /fare sıçan gibidir

2) Maymun köpeğe benzemez /maymun köpek gibi değildir

Thank you for the replies.

But they raise another question......why does ´gibi´ get the -dir suffix added?

The suffix dir is the equivalent of is in english, especially for third singular person.

Examples: o iyidir, o kütüdür, o faydalıdır, o soğuktur, etc...

Is this a rule for "gibi" to add -dir or you add it only if you want ?

No it´s not particular to gibi, but I think it´s grammatically more correct to add the dir suffix for formal speaking and writing..
And you can write gibi without dir.

Ex: çocuk gibi...

We have rules. 

"gibi" can either be used as "edat" (particle) or "zarf" (adverb). When used as "edat" it takes -dir:

hayat bir film gibidir - life is like a movie

When used as "zarf" it takes no suffix:

çılgın gibi koştu - he ran like crazy

it has nothing to do with "gibi" but because you are stating a fact here.

it is known to everyone, or that it is a fact, that mouse is like rat, so you should say "fare sıçan gibidir", just like you should say "kar beyaz renklidir" or "fare bir hayvandır"

-dir suffix is not always necessary.

"o iyi" or even "iyi"= she/he/it is good

"o kötü" or even  "kötü" = she/he/it is bad

"o faydalı" or even "faydalı" = she/he/it is useful

"o soğuk" or even "soğuk" = she/he/it is cold

-dir is usually preferred when you are being informative to the addressee (you think they don´t know or are not aware of what you state, etc.).

-dir suffix comes from turur (means stands). Over time, it has become what it is now.

The following is a quote from E-prime (English language without "to be") page:

Some languages already treat equivalents of the verb "to be" differently without obvious benefits to their speakers. For instance, Arabic, like Russian, lacks a verb form of "to be" in the present tense. If one wanted to assert, in Arabic, that an apple looks red, one would not literally say "the apple is red", but "the apple red". In other words, speakers can communicate the verb form of "to be", with its semantic advantages and disadvantages, even without the existence of the word itself.

In Turkish "to be" is equivalent to suffixes for present tense. But third single person usually behaves like Russian or Arabic in present tense form. Or sometimes like English (using -dir suffix) as described above.