1. On the way home I met my old friend.

What is the part of speech of 'home'?
Is it an adverb? Then, what does 'home' modify?

2. Probably tomorrow he'll come back.
3. Maybe tomorrow he'll come back.

What is the difference between 'probably' and 'maybe'?

4. He got there in safety.
5. He got there with safety.

Which one is right between #4 and #5?
Do #2 and #3 have different meaning?

6. He walked through the shallow river with care/ withour care.
(Is this sentence grammatical?)

1. On the way home I met my old friend.

What is the part of speech of 'home'?
Is it an adverb? Then, what does 'home' modify? Yes, all that is adverbial, including "home" -- modifying the verb "met." The adverb phrase/words are telling when and where "met" happened.

2. Probably tomorrow he'll come back.
3. Maybe tomorrow he'll come back.

What is the difference between 'probably' and 'maybe'?
no difference

4. He got there in safety.
5. He got there with safety.

Which one is right between #4 and #5?
Do #2 and #3 have different meaning?
Neither 4 nor 5 is correct. You should say, "He got there safely." Sentences 2 and 3 have the same meaning.

6. He walked through the shallow river with care/ withour care.
(Is this sentence grammatical?)
It'd be better to say, "He walked through the shallow river carefully." (or "carelessly")

1. In the sentence "On the way home I met my old friend," 'home' is not an adverb but a noun. It functions as the object of the preposition 'on' and identifies the destination or endpoint of the action of going.

2. 'Probably' and 'maybe' are both adverbs that express uncertainty about a future event. However, there is a slight difference in their usage and degree of certainty. 'Probably' indicates a higher likelihood or probability, suggesting that something is likely to happen. On the other hand, 'maybe' indicates a lower level of certainty or possibility, implying that something might happen but with less assurance.

4. The sentence "He got there in safety" is the correct form. 'In safety' is an idiomatic expression that means arriving or reaching a place without any harm or danger.

5. The sentence "He got there with safety" is not grammatically correct. The preposition 'with' is typically used to indicate the means or manner of doing something, but it is not the appropriate word to use in this context.

6. Yes, the sentence "He walked through the shallow river with care/without care" is grammatically correct. The choice between 'with care' and 'without care' depends on the intention or attitude of the subject towards the action of walking through the river. Using 'with care' implies being cautious and taking precautions, while 'without care' suggests a lack of concern or attention to potential risks.