First of all salams and thank you for referring to this website to get the answers to your questions.
The first thing that must be noted is that being tamed and trained have no affect on the rulings regarding these animals. The only exceptions we have in our hadiths have to do with hunting and guard dogs. The answers to all of your questions on cats can be found in Question 697 (website: 745). Here we will address your questions on dogs.
It is not haram to touch dogs even though they are najis, and the only way one’s hand becomes najis by touching them is if at least one of the two (the hand or the dog) has enough moisture for it to be transferred to other. But if both surfaces are dry or only damp to the extent that no moisture is exchanged and transferred from one to the other, the hand or whatever else that has come in contact with the dog will not become najis.
Therefore, the only way your body or clothing or anything else will become najis is for you to know for certain that it has come in contact with the dog while one or both of had enough moisture for the najasah to transfer (thus, if you aren't sure if it came in contact or not, or you know it did but aren't sure if any of the two were moist or not, or if you know they were moist but don’t know if moisture was sufficient enough for the transfer of najasah, in all these cases, what has come in contact with the dog is deemed tahir and isn't considered najis).
Also, regarding praying with clothing that has come in contact with a dog, you can refer to Question 2828 (website: 3055).
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