1. Being ill, he played football.
2. Although he was ill, he played football.
( #1 is incorrect. #2 is right. We cannot change #2 into #1. Is that right? Where can we find the explanation about that?)
3. Being ill, he couldn't go to school.
4. Because he was ill, he couldn't go to school.
(#3 is right. #4 is right. #4 can be changed into #3. Is that right? Where <grammar book or grammar site> can I find the information about that? Thank you.)
2 is correct. 1 is ambiguous.
3 and 4 are interchangeable, yes.
These websites will help you find grammatical information (explanations and examples). For your current question, look up their information on participle phrases, especially introductory participle phrases.
http://www.englishclub.com/grammar/
http://grammar.ccc.commnet.edu/grammar/index2.htm
http://www.chompchomp.com/terms.htm
Yes, you are correct. The sentence "Being ill, he played football" (#1) is grammatically incorrect, while "Although he was ill, he played football" (#2) is the correct construction. The reason for this is that in English, the phrase "being ill" is not typically used at the beginning of a sentence to indicate a cause or reason. Instead, we usually use subordinating conjunctions like "although," "because," or "since" to introduce subordinate clauses that provide reasons or explanations.
Similarly, the sentences "Being ill, he couldn't go to school" (#3) and "Because he was ill, he couldn't go to school" (#4) are both correct. In this case, both "being ill" and "because he was ill" are acceptable ways to express the cause or reason for not being able to go to school.
If you are looking for a grammar book or grammar site to find more information about these grammatical constructions, you can refer to reliable grammar books like "The Elements of Style" by William Strunk Jr. and E.B. White, "A Comprehensive Grammar of the English Language" by Randolph Quirk et al., or websites like Grammarly, the Purdue Online Writing Lab (OWL), or Cambridge Dictionary's Grammar section. These resources should provide you with detailed explanations and examples of various grammar rules and constructions.