Do all labs carry the gene of a chocolate lab and black lab and a yellow lab?

No, all labs do not carry genes for all lab colors.

Black is dominant to chocolate, so a black dog might be black with a hidden chocolate gene (represented Bb) or with two black genes (BB). Only one B is needed to make black, two b's are needed to make a chocolate. To be black, the dog must also have at least one gene for expression the dark coat (represented as an E). A dog without a gene for expression of the dark coat would be e/e, and be a yellow.

So each lab has two pairs (simplifying) of genes controlling color. A dog's genes can be described with the pairs of letters.

Black dog = genes of,

BB + EE (will never have a chocolate or yellow puppy), or

Bb + EE, (never a yellow puppy), or

BB + Ee, (never a chocolate), or

Bb + Ee (this would be black and carrier of all colors, the others do not carry all colors, the colors in the dogs puppies would depend on the other parent, except that there is a chance of black no matter what the other parent carries).

Chocolate dog =

bb + Ee, or

bb + EE (no black in the chocolate dog's genes, if there was it would be a black dog)

Yellow dog=

BB + ee,

bb + ee (will have liver-colored nose),

or Bb + ee (the last is yellow and another carrier of all colors)