Hi Robert, there are many labs in the Midwest that do pesticide residue testing. Some are limited to specific pesticides or require a certain number of samples to be submitted, so check with the lab before you send anything in. An internet search turned up Alliance Analytical Labs, which is close to Grand Rapids. I don't have any experience using them, but the website is user-friendly and might be a good place to start. Make sure to ask if they can test for the specific active ingredient in the product that was sprayed. 

I haven't found an insecticide called Tran Star, but there is one called TriStar that is labelled for turf, so I'm guessing that's what was applied on your neighbor's yard. To know for sure, you will have to ask your neighbor or the company for the product name or label. If the product was TriStar, the active ingredient is acetamiprid, so make sure to ask the lab if they are able to test for acetamiprid residue before you send a sample in. I do not know of any labs nearby that test pesticide soil contamination; however, it is unlikely that the pesticide would be at detectable levels in your soil if it drifted over from a neighbor. In addition, acetamiprid tends to break down quickly in the environment, so residual soil contamination is unlikely. I would not recommend testing the soil unless there was a spill or direct overapplication in your soil.

As far as chemical effects on humans and pets, reference the TriStar Safety Data Sheet (SDS) for all that information. Each pesticide registered with the EPA must have an SDS with chemical contact, first aid, toxicological, ecological, and other information. (If you find out TriStar is not the product that was applied, simply google search the actual product name and add "SDS" behind it, and it will pop up.) I recommend you read the full sheet, but here I've included the toxicological information section of the TriStar SDS:

"Likely Routes of Exposure: Dermal, inhalation
Symptoms of Exposure:
Eye Contact: Not irritating based on toxicity studies.
Skin Contact: Not toxic and not irritating based on toxicity studies.
Ingestion: Slightly toxic if ingested based on toxicity studies.
Inhalation: Low inhalation toxicity based on toxicity studies.
Delayed, immediate and chronic effects of exposure: None expected.
...
Eye Irritation: Rabbit: Not irritating
Skin Irritation: Rabbit: Not irritating
Skin Sensitization: Not a contact sensitizer in guinea pigs following repeated skin exposure.
Subchronic (Target Organ) Effects: Acetamiprid does not appear to have specific target organ toxicity.
Carcinogenicity / Chronic Health Effects: No data available.
Reproductive Toxicity: There is no evidence of any teratogenic effects due to treatment with acetamiprid.
Developmental Toxicity: There are no signs of developmental toxicity were observed.
Genotoxicity: No data available.
Assessment Carcinogenicity: None listed with ACGIH, IARC, NTP or OSHA."