(Circled in the image)

The doctor and teacher were women. Hence, the ‘Michael’ the dead woman had addressed prior to being shot could only be the lone man, the lawyer.

Alternate possibility: The doctor and teacher were already known to the detective – he knew that neither of them were named ‘Michael’, so the ‘MIchael’ in question could only have been the lawyer. 

Pirates of the Caribbean (Pi-rats-…) 

Jigsaw puzzle / pieces

The picture clue makes the answer too obvious!

3

Hint / clue: Binary representation of numbers, with balls representing 1’s and rings representing 0’s.
001 = 1,
010 = 2,
100 = 4,
011 = 3

ALASKA, one of the fifty states of the USA.

1. Split the bottles into sets of 3.
2. Take the first set. (Say, bottles A, B, and C) Mix a small quantity from each of the three bottles and feed the mix to one of the mice.
2.1. If the mouse died, it means one of the bottles is the poisoned one.
2.1.1. Since A, B or C is poisoned,  feed the other mouse with a small quantity from one of the bottles, say, A.
2.1.1.1 If the mouse died, bottle A is poisoned.
2.1.1.2 If the mouse remained alive, it means bottle A  is safe. Keep it aside and feed the mouse from one of the remaining bottles.
2.1.1.2.1 If the mouse died, it means the bottle selected is poisoned.
2.1.1.2.2 Otherwise, the remaining bottle is the one poisoned.

2.2. If the first mouse  remained alive after consuming the mix, it means that all bottles in the first set are safe.

We now have two more sets of bottles and BOTH the mice alive. Hence, repeat the same steps as above (2 –  2.1.1.2.2) with the second set of bottles. We can determine either the poisoned bottle among Set 2 OR that all bottles in Set 2 also are safe, similar to the case of Set 1.

If both the mice remained alive after second round, repeat the same with the third and final set of bottles.

You will either identify the poisoned bottle OR learn that you have been tricked to believe that one of the bottles is poisoned. 🙂 

Two WRONGs do not make a RIGHT.

Either one OR both references to ‘Thursday‘ could point to the name of a horse rather than the day, so that the tourist could have ridden on Thursday, the horse during one or both of the trips, and if only one of the trips was on the horse ‘Thursday‘, the other could be on the day, Thursday’.