1. He is good at playing billiards.

2. He is good in playing the billiards.
(Can we use 'in' instead of 'at'?)

3. It's the first play-off game, and I'll be the catcher.

(What is the meaning of 'the first play-off game'?)

4. I am disappointed because this is a very important game for me.

5. I am disappointed because this is a very important game to me.

(Can we use both 'for' and 'to' in the sentences above?)

1 is correct; 2 is incorrect. (Don't use "in" in this situation.)

3 is correct. A play-off game is part of a series of games after the regular season is over; these games are played to determine who will be first in the whole league or district. Here are several websites about the basketball playoffs going on in the US right now:
http://www.google.com/search?q=nba+playoffs&rls=com.microsoft:en-us:IE-SearchBox&ie=UTF-8&oe=UTF-8&sourceid=ie7&rlz=1I7GGIE_en

4 and 5 are both correct; 5 is probably used more often.

1. Yes, we can use "in" instead of "at" in sentence 2. Both "at" and "in" are commonly used to describe proficiency in a particular activity or sport.

3. "The first play-off game" refers to the initial or inaugural game in a series of play-offs. Play-offs are a series of games or matches played within a sports league to determine the champion or top performers.

4. Yes, we can use "for" in sentence 4. It indicates that the game holds significance or importance in the context of the speaker.

5. Yes, we can use "to" in sentence 5. It also conveys that the game is significant or holds importance for the speaker. Both "for" and "to" can be used interchangeably in this context.

1. Yes, we can use 'in' instead of 'at' in the sentence "He is good in playing billiards." However, the more commonly used preposition to describe proficiency or skill is 'at'. So, "He is good at playing billiards" is more commonly heard or seen.

2. "It's the first play-off game, and I'll be the catcher." In sports, particularly in team tournaments or leagues, a play-off game refers to a competition or match that takes place after the regular season to determine the final standings or winners. It usually involves teams competing against each other in a series of matches, with the ultimate aim of reaching the finals and becoming the champion. So, when someone refers to the "first play-off game," they are indicating that it is the initial game in the play-off series.

3. No, we cannot use both 'for' and 'to' in the sentences you provided. In sentence 4, the more appropriate preposition is 'for', as it indicates the reason or cause of disappointment. "I am disappointed because this is a very important game for me." Here, the importance of the game is the reason for their disappointment.

In sentence 5, 'to' is the suitable preposition to use. "I am disappointed because this is a very important game to me." It expresses that the game holds personal significance or meaning for the person expressing the disappointment.