1. Do we have any consulting firms in USA which are helping to get teaching job? just like, how engineers get job through consulting firms, if any company is ready to offer job.
You have to explore local universities, their present workforce needs and walk your resume into select departments. Academic candidates benefit from posting on Indeed website when they clearly indicate locations and subject areas of interest.
2. Anybody obtained Postdoctoral fellowship, when they applied on H4 visa?
You can study but you can't work on H-4. A fellowship is a paid position. As Kumar said, you qualify for cap exempt H1B and can start working at most universities once approved. However, J1 is an inferior option to do this due to set time limits on this type of visa.
3. if so, converting visa status from H4 to J1, is University is taking care?
Your conversion should be from H4 to H1B and not J1. Yes, university will take care of this.
4. After completion of Post doctoral fellowship, I am eligible to work on J1 visa or visa status is going to change as H4.
You are not eligible to seek employment on J1 but only move from one fellowship to the next. Not a good option.
5. What other options are available to continue my career in USA? Kindly let me know.
Find a position, teaching or research. Apply, get selected. Request university to process H1B. As Immigeek stated, based on your performance, publications etc you will be eligible for EB1 green card category reserved for 'outstanding researchers / professors' in a few years.