>>4 A humane trap won't injure the animal, and it will allow you to carry / transport them without sustaining injuries yourself. If the bars are big enough for them to poke their arms through, though, you should drape a thick material (like leather) around it or put cardboard between yourself and it.
It's not hard to tell if a cat is stray or not. If the cat has no collar, it is most likely a stray; responsible cat owners understand the importance of a cat collar. If the cat is excessively dirty, heavily matted, bloody, very thin, etc. it is obviously not being taken care of and needs to be brought to a shelter even if it has a collar. Let them sort it out.
If you're not sure, you can take care of the cat for a few days and post Found Cat signs around the area. If nobody picks it up, take it to the shelter.
Most people microchip their pets these days, so the shelter will be able to check where the pet came from. Even if the individual didn't microchip them, the shelter or breeder they bought it from most likely did and has their information in it.
What you do need to be wary of is feral cats. These cats usually have no collars but are very clean. They're often larger than house cats, more muscular, and much more brave around humans. These are cats that grew up on the streets without ever being tamed. Most shelters won't take them, and you're best contacting a specialist to get them spayed, which will help control the feral population. Feral kittens should be taken to a shelter. You can find a list of organizations to help with feral cats in the
US here.