@Madala, Hanumantharao Based on your requirements, I believe you just looking for conditional statements in your logic apps.
- Incase there are no files in source path, no need to proceed further.
For this, you could just check if the returned array from the List files in folder action is empty. You can use the empty function in a condition and if true, call the terminate action.
- If files are present in source folder, but not complete files(part files).. how to handle these files with exception handling?
Here you can filter out the list of files based on something like the filename (assuming the filename is different for part files in your case).
- If correct files are present in source folder, during connection to destination folder failed. How to catch this exception and send an email?
This is where you would use run after. For this, you would need a parallel branch that only runs when the previous failed.

