1. He was excited to hear the news.

2. He was excited that he heard the news.
(Are both the same? Or does #1 mean #2-2 as follows?)
2-2. He was excited that he would hear the news.

3. He was excited to watch the game.
4. He was excited that he watch the game.
5. He was excited that he would watch the game.
(Are the three sentences the same in meaning? Does #3 mean #4 or #5?)

1 and 2 mean the same thing, yes.

2-2 is not the same ... "would" implies that it's something that might happen in the future ... or might not.

3 is correct on its own.
4 is incorrect.
5 is somewhat correct, but has more of a "possibility" meaning, just like 2-2 above.

1. The sentence "He was excited to hear the news" indicates that the subject of the sentence was excited when they heard the news.

2. The sentence "He was excited that he heard the news" also conveys excitement based on the fact that the subject heard the news. However, there is a subtle difference in tense between this sentence and the first one. The use of "heard" suggests that the event of hearing the news has already taken place.

2-2. The sentence "He was excited that he would hear the news" implies that the person was excited about the expectation of hearing the news in the future. The use of "would" indicates a future action.

In summary, the first two sentences (1 and 2) convey the same meaning but with a slight difference in tense. Sentence 2-2 (the modified version of 2) introduces the expectation of hearing the news in the future.

Moving on to the next set of sentences:

3. The sentence "He was excited to watch the game" indicates that the person was excited to observe or view the game.

4. The sentence "He was excited that he watch the game" contains an error in grammar. The verb "watch" should be in the past tense, so it should be "He was excited that he watched the game."

5. The sentence "He was excited that he would watch the game" suggests that the person was excited about the anticipation of watching the game in the future. The use of "would" indicates a future action.

In this case, sentences 3 and 5 convey similar meanings. Sentence 4, once corrected, would also convey the same meaning as 3 and 5.

Please note that context and additional information might be required to fully understand the intended meaning of these sentences.