1. There are many taekwondo centers in London, with lots of students.

2. There are many taekwondo centers with lots of students in London.

3. There are many taekwondo centers in London with lots of students.

(Are they all the same? What is the part of speech of 'with lots of students' in Sentence 1? Do we have to put a comma before 'with lots of students' as in Sentence 1?)

The clearest sentence is number 3 -- and it's also punctuated correctly.

"with lots" is a prepositional phrase used as an adjective modifying "centers."

"of students" is a prepositional phrase used as an adjective modifying "lots."

1. "There are many taekwondo centers in London, with lots of students." - In this sentence, the phrase "with lots of students" is functioning as an adverbial phrase. It provides additional information about the taekwondo centers by indicating that they have a large number of students.

2. "There are many taekwondo centers with lots of students in London." - This sentence has the same meaning as the previous one (#1), but it rearranges the phrase "with lots of students" to the end of the sentence. The part of speech remains the same; it is still functioning as an adverbial phrase.

3. "There are many taekwondo centers in London with lots of students." - This sentence is similar to the others, but it removes the comma between "London" and "with lots of students." The phrase "with lots of students" is still an adverbial phrase providing information about the taekwondo centers.

All three sentences convey the same meaning, but differ in their word order and punctuation. The choice of which version to use depends on personal style and preference. The use of a comma before "with lots of students" in Sentence 1 is optional, as it can be considered a non-essential phrase that adds additional information.