How sensitive is the breakaway feature? Does it snap constantly in the first week of use?

No...I found it much stronger but my cat is indoors...the collar is loose and still a bit hard to press it open. She has plenty of freedom to move.I strongly recommend you have her chipped for safety if she's an outdoor cat. My other cat was an outdoor cat but I let him out in the daytime, kept him in at night, he lived a lot longer that way.

Will this fit a 6 month old and 5.3 lbs cat? I'm having a hard time looking.

The collar has several clasp connectors. I inspected my other 3 collars and if you keep playing with it you will see it is adjustable for longer and shorter lengths. It may take a few minutes to get the right fit. My cat only weighs 8-10 lbs. and is considered smaller that the average feline. I keep her collar on the loose side but squeezing the clasp open is very hard to release. It is a very tight clasp...so I just pull it over her head. Still it is sturdy and a great bargain for 4 collars I think.

What is material made form? my cat is sensitive with nylon

This collar design works for me. While I still could not open the latch I was able to get the collar of my cat and off by gently but quickly pulling it down over her face. She is a smaller cat and about 15 years old. Since this is yet another item made in China without the products ingredients listed I am not sure if there is nylon in it. Izzy is a long haired cat which gives her some protection from direct contact. The collar appears to a have a plastic feel skin side which could very be well contain nylon. I found it to be a good deal, they are affordable, adjustable, colorful, and the bell is a plus. Perhaps the company can get your question from the manufacturers in China.

Will this fit an teacup puppy?

I really don't know because I bought them for a small cat. The description should give a measurement for the collar and then you could measure your dog's neck.