1. Smells occur ONLY after a rain and are very string OUTSIDE on both ends of the house.

"Both ends of the house" are you referring to the front of the house and the back of the house? Is the front of the house where the main drains to the sewer?

2. Odor does come inside and seems to be most highly concentrated in a second floor closet, not near any vent or plumbing. There is no air duct into this space.

Baffling.

3. NO smell in attic

ok

4. Partial basement then crawl space. No leaking in crawl or smell in crawl. Everything appears OK.
5. NO SMELL coming from any drains or in any bathrooms.

Makes sense, each fixture has a p trap

6. Ran water down 4 roof vents. One is blocked. It services one bathroom. This bathroom never smells. Plumbers cant get the snake to go thru. May try high pressure water treatment.

Probably not related.

7. Worst smell is at the opposite end of the house.

As with #1 above, is the opposite end of the house where your main goes to the sewer? Is there another house at this end of the house?? Have they had their septic converted recently also?

8. Reminder, we never had a problem while on septic system.

What time of year was the system converted? My first thought is maybe you have a pipe separation that occurred after install in hard, dry ground and separates/separated when it rains and the ground swells. I don't think it is a coincidence that after the convert you smell gas - something is wrong with the conversion. The smell you get inside is hard to fathom not knowing the layout of your house. Seems very strange that a interior (?) room on a second floor with no plumbing, HVAC vents (true?) or windows would smell like sewer gas.

Referring to "Both Ends" of my house, think of ranch style. Strong at the side where the drain line exits the house and where we connected to the new sewer line. Other end, is the opposite side, the furthest away from this connect.

The installation occurred in spring, however Indianapolis had a drought over the summer. The ground swell idea is interesting. Sewer pipe was laid down on a bed of gravel. After the inspection, more gravel was added then soil.

Affirmative on the closet that smells, interior, no windows, vents or plumbing. We are baffled. Going to do the smoke test.

I think this is a simple matter of who did the sewer hookup and what is their warranty/liability. Basically, it seems as though it should be someone else's problem to fix and not yours. You didn't lay the pipe yourself.

If you do the smoke test, please post back with your results. Everyone loves a mystery!!!

And, was your septic also "decommissioned" properly? Punched out and filled with gravel?