No, Talstar P is an insecticide that does a great job killing insects, but it will not kill mice.
Yes, Talstar P can be used for fire ant control in the type of sprayer you described, but you will also need to do mound drenching. According to the Talstar P product label, "Fire Ant Mound control is optimized by combining broadcast applications that will control foraging workers and newly mated fly-in queens with mound drenches that will control existing colonies. If the soil is not moist, then it is important to irrigate before application or use a high volume application. Apply broadcast treatments at 0.5 to 1 fluid oz. per 1,000 square feet. Use enough finished volume to penetrate thatch or sod. Treat mounds by applying 1 oz of Talstar P per mound in 1 to 2 gallons water by sprinkling the mound until it is wet and treat 3 feet out around the mound. Use the higher volume for mounds larger than 12”. Treat mounds with sufficient force to break their apex and allow the insecticide solution to flow into the ant tunnels. For best results, apply in cool weather (65 - 80F) or in early morning or late evening hours."
Talstar P (Talstar One) after dilution is only strong enough to kill small insects. Your puppy would have to eat loads of treated grass for Talstar it to harm it. Be sure to keep the puppy out of the treated area until dried.
Use a 0.06% dilution of Talstar P (1 fluid oz. per gallon of water) for bedbug control. A thorough application should be made to crack and crevices where evidence of bed bugs occurs. This includes bed frames, box springs, inside empty dressers and clothes closets and carpet edges,high and low wall moldings and wallpaper edges. Do not use this product on bed linens, pillows, mattresses or clothes. Remove all clothes and other articles from dressers or clothes closets before application. Allow all treated areas to thoroughly dry before use. Not recommended for use as sole protection against bedbugs. If evidence of bedbugs is found in/on mattresses, use products approved for this use such as Bedlam aerosol and Protect-A-Bed mattress, boxspring and pillow encasements
While we do not recommend to apply Talstar P (Talstar One) at a higher rate than 1oz, your lawn should be just fine. Please keep in mind that it's also recommended to apply during cooler times of the day.
Talstar P (Talstar One) is labeled to be applied to cracks and crevices indoors. Mopping should not be an issue, as Talstar is not labeled to be applied as a broadcast over the floor. If you were to mop along the edges of the wall where Talstar was applied, it would take it up.
It should not harm anything to apply Talstar at this rate one time. We recommend that you wait for 30 days outside (90 days inside) to reapply at the rate of 0.5-1 oz per gallon of water.