1. I have hunger to go home now.

2. I have a hunger to go home now.
3. I am dying to go home now.
4. I am looking forward to go home now.
5. I anticipate going home now.
6. I expect to go home now.

(Are they grammatical? Which one is correct, #1 or #2? Do all the sentences have the similar meaning?)

1. is not grammatical.

2 is grammatical but not used very often.

3. is commonly used.

4 is ok if we say "I am looking forward to GOING home now."

5 and 6 are grammatical and common.

All of the sentences you provided are grammatically correct, but they have slightly different meanings and convey different levels of intensity.

1. "I have hunger to go home now." - This sentence is grammatically correct, but it sounds a bit unusual or overly direct. Normally, when expressing a strong desire to go home, it is more natural to say "I am hungry to go home now" or "I am eager to go home now."

2. "I have a hunger to go home now." - This sentence is also grammatically correct and conveys a strong desire to go home. The use of "a hunger" instead of just "hunger" adds a slightly poetic or figurative touch, suggesting a compelling need or longing to return home.

3. "I am dying to go home now." - This sentence expresses an intense desire to go home. It uses dramatic language to convey extreme impatience or eagerness. It is an idiomatic expression or turn of phrase and doesn't indicate an actual medical condition.

4. "I am looking forward to go home now." - This sentence is grammatically incorrect. Instead, it should be phrased as "I am looking forward to going home now." This sentence indicates an eager anticipation or excitement about going home in the near future.

5. "I anticipate going home now." - This sentence is grammatically correct and conveys a sense of expectation or anticipation regarding going home. It suggests a more measured or deliberate state of mind compared to the previous expressions.

6. "I expect to go home now." - This sentence is grammatically correct and implies a certain degree of certainty or reliance upon going home. It suggests that the speaker has made plans or has reasons to believe that they will be able to return home soon.

While all of these sentences express a desire or longing to go home, the intensity varies, with sentences 3, 4, and 6 being slightly less intense compared to sentences 1, 2, and 5. Sentence 5 has a more formal tone compared to the others. The choice would depend on the desired level of intensity and formality in conveying the sentiment.