Praveen,
I think you are eligible to appear in both the modules of the CS Executive examination in June 2013 examination is not that correct. Mind you, as per the latest news, the revised syllabus has 7 papers to clear as against 6 papers in the existing scheme. My suggestion to you is to apply for both modules as it is relatively easy and that too you are from CA background (practicing CA) and finish the course at the very earliest.
On your query on part (a) yes, you ought to write any one paper from test series 3 or test series 4 or test series 5 for each subject and submit to the Institute within the cut-off date.
Note: Most students do not submit their postal paper but still apply for examination and ICSI permits them to appear in the examination. My suggestion is to get started to write the test paper at the earliest as this will provide you with a macro view of the syllabus.
Yes, I presume you have fair computer knowledge and it is advisable for you to take up online test.
Now on to your query of part (b) ICSI latest study material should be sufficient and complement it with guideline answers of 5-6 attempts.
Hope the above information is useful for you to take an informed decision.
Best of luck
Sathish