Are the tenses used in the sentence correct?

From http://dilbert.com/strip/2015-07-31:

I'm going to call you Carlos from now on. And it would help if you grew a beard and walked with a limp.

Shouldn't it be grow a beard and walk with a limp in this sentence?

The tenses of the verbs in the second sentence are conditional. In English there are various types of conditional tenses depending on the nature of the condition.

In this case the condition is what http://www.edufind.com/english-grammar/conditional/ calls a Type 2 Conditional:

The type 2 conditional is used to refer to a time that is now or any time, and a situation that is unreal. These sentences are not based on fact. The type 2 conditional is used to refer to a hypothetical condition and its probable result. In type 2 conditional sentences, the if clause uses the simple past, and the main clause uses the present conditional.

"it would help" is the main clause and is in the present conditional tense.

"if you grew a beard and walked with a limp." is the if clause and uses the past tense.

"it will help if you grow a beard and walk with a limp." (a type 1 conditional) could be used but requires the main clause to be in the simple future tense and is less speculative, which would be less appropriate as we have no indication that Dilbert intends to grow a beard or walk with a limp.

It's because it's hypothetical ("it would help if... ").

With hypothetical conditions (or 'unreal') you always use the past tense in the if clause:

If I had a million dollars, I would buy you a house. (but I don't have a million dollars and so this situation is unlikely)

With ' real' conditions, you use the present tense in the if clause:

If I go to the store, I'll buy some milk. (this is a real situation that will probably happen)

Rendered in upper- and lowercase letters instead of all caps, the pointy-haired boss's words in the last panel of the Dilbert cartoon that you refer to are as follows:

So... I'm going to call you Carlos from now on. And it would help if you grew a beard and walked with a limp.

The speech bubble breaks neatly into a first sentence in which the boss explains what he is going to do from now on, and a second sentence in which he explains what it would be helpful for Dilbert to do, if Dilbert would agree to do it. The first sentence is expressed in a forward-looking form of the present tense ("I'm going to ..."); the second sentence is expressed entirely in conditional form, with the consequence—"it would help"—tied to two parallel conditions:

• if you grew a beard

and

• [if you] walked with a limp.

The wording is perfectly grammatical as written. If the boss had cast the conditions as "if you grow a beard" and "if you walk with a limp," the consequence would logically become "it will help," since the consequence (considered as a temporal event) must follow the fulfillment of the conditions, and if the conditions to be fulfilled are in the present, the consequence must be in the future.

Yes, they are correct.

Grew and walked are in the subjunctive mood.

I found a nice explanation of this at http://dictionary.reference.com/browse/subjunctive:

Subjunctive: a mood or mode of the verb that may be used for subjective, doubtful, hypothetical, or grammatically subordinate statements.

That link has some more helpful information and I recommend that you take a look at the whole entry.