Hello Adarsh,
1. Institute might take 2 weeks to 2 months to issue Letter of Good Standing. Keep on calling and reminding them at the numbers available.
2. There ae many occupations in which we can work. So first you need to identify occupation from SOL. If your occupation is assessed by ICAA or CPA or IPA institute, then minimum of 7 bands in Academic IELTS is required. And if you want to apply as internal auditor then Vetassess will assess your documents and it doesn't require IELTS. However for PR Visa, you require minimum of 6 bands in General or Academic IELTS. So in any case you have to do IELTS. Just verify which module you will be needing General or Academic.
3. There is no need of migration agent. Australian proceesses are very well documented and www.expatforum.com is a very useful website with lots of helpful advisors. People are applying without migration agent because migration agent just helps in scanning and uploading docs and maintaining timelines. He can't help your file move faster or can't even influence it any way. Also, to manage all the required documents will be your responsibility and they will not run here and there for you. Better do it yourself. If you can study CA from ICAI, you can easily crack migration process.
Note: Entire migration process might take 6 months to 1 year.
Best of Luck.