• Is singular they widely accepted in English speaking countries such as America and Britain?

In the 19th century, singular they became stigmatized. It wasn't until the 1960s that this began to change, when feminist linguists began to promote singular they. But, within the past few decades, it has become more and more accepted. At this point, yes, many people accept it but it's still a little controversial. Some (prescriptivist) grammar books will still tell you it's ungrammatical. But things are still changing, and singular they is becoming more and more used and accepted.

• Is it formal enough to be used in formal situations, such as a public speech?

Again, there is some controversy about how formal it is. Some people, myself included, use singular they in formal situations, such as essays and speeches. However, I think I might be in the minority. According to research, singular they isn't used much in academic writing:

In contrast, Adami’s (2009) findings seem to confirm Baranowski’s (2002) suggestion that he is still prevalent in more formal contexts. She investigated the use of generic pronouns in several academic corpora and found that although generic he still appeared to be the preferred choice for singular generic reference, its use had decreased substantially from the 1960s (Adami 2009, 291- 292). Interestingly, she suggests that the “gap” left by the decrease of he was in British English filled by the increased use of he or she, while Americans may prefer other strategies such as pronoun avoidance and pluralisation (ibid.). Use of singular they had not increased significantly, however, and the pronoun was equally infrequent in both varieties which, according to Adami (2009, 292-295), seems to suggest that the proscription of they still affects academic writers and editors.
_{Generic personal pronouns in New Zealand newspaper English}

If you're using a style guide, then it may have its own rules. Wikipedia lists some examples here.

• Is it casual enough to be used in informal situations, such as gossiping among friends?

Yes, and you will find native speakers using it. This is true for American English (my dialect), British English, Australian/NZ English and many others (as my research indicates). It's used by educated and uneducated alike.