As far as I know no IDE sends any kind of alerts to a screen reader when you enter and exit methods. You need to read the current line of the text and derive this knowledge from what is happening in the code.
Eclipse is one of the most accessible IDEs. However, as far as I know, you can no longer use it for android development.
You may also want to try visual studio code and visual studio community addition.
<snip 8. The NVDA should be able to read out the Quick Fix options in Android Studio.
9. The NVDA should be able to read out the possible suggestions given by Android Studio while I do programming.
10. The NVDA should alert me while I do my debugging process using breakpoints etc.
11. The NVDA should alert me about the launching of app or about the failures while launching process.
12. The NVDA should properly read the errors that occur while building the program.
PL] For the above points, please raise issues on both nvda and on android studio. This is because it is hard to determine where the problems are.