1. I hope to succeed.

2. I hope that I will succeed.
3. I hope my success.
4. I hope for my success.
5. I hope for success.

(Are they all the same and grammatical?)

yes they are the same but not grammatical. number three is wrong

Anthony is correct.

All the sentences you provided are grammatically correct, but they have slightly different meanings.

1. "I hope to succeed." - This sentence expresses the speaker's desire or expectation of achieving success in general. It implies that the speaker has a goal or ambition they are striving for.

2. "I hope that I will succeed." - This sentence conveys the same meaning as the first one but adds the word "that" to introduce a subordinate clause. The clause "that I will succeed" provides more clarity and explicitly states what the speaker is hoping for.

3. "I hope my success." - This sentence, while grammatically correct, is less commonly used. It implies that the speaker wants to possess or own their success, almost treating it as a tangible object. It might be slightly confusing or unusual for most listeners or readers.

4. "I hope for my success." - This sentence suggests that the speaker is anticipating or wishing for their own success. It implies a personal attachment to the outcome, as if the speaker is waiting for their success to happen.

5. "I hope for success." - This sentence expresses the speaker's general desire for success, but without explicitly mentioning personal involvement or specific goals. It can be interpreted more broadly, implying a wish for success in a particular endeavor or for others as well.

In summary, while all the sentences are grammatically correct, they have different nuances and slightly varying meanings. The intended meaning and context should be considered when choosing the most appropriate sentence to convey your thoughts.