You seem to be confused about so many things I don't even know where to start.
Degree to which a weak acid (and lactic is a weak acid) is protonated in a solution is a function of the pH. pH can be result of the presence of the weak acid itself, it can be also forced on the solution by the presence of other acids or bases. In general the lower the pH, the more protonated the acid is, but it also depends on the acid identity (or, more precisely on its dissociation constant).
Lactic acid dissociates in water solutions. Lactic acid - when solid* - is fully protonated. When dissolved - it partially gets rid of these protons. Only partially, as it is a weak acid, that is not dissociating 100%. How much is dissociated depends on the concentration (and dissociation constant, but for a given acid it is - as the name implies - always the same). For example, if you dissolve 1g of lactic acid in 100 mL, around 3% will be dissociated (so not protonated), rest is protonated.
Lactic acid molecules are neutral. When they dissociate
they produce H+
anion. The neutral form on the left is the protonated one.
dl means racemic mixture. Lactic acid has two optical isomers (its molecules can take two forms, each of them rotates polarized light in a different direction). One of these isomers is designated l, the other d. dl means both isomers are present in the mixture is equal amounts. (+) in this context means exactly the same thing as d. Note that dl and DL are not equivalent (even if they refer to the same thing). l(+) doesn't make sens, L(+) can be correct.
As (+) in the name refers to the polarized light rotation, in the context of naming it has nothing to do with the acid protonation.
*Solid or liquid, as it depends. If there is only one optical isomer, lactic acid melts in much higher temperature than the mixture. Mixture is a liquid at room temperature, while single isomers melt at 53°C.