Are those bleaching trays safe?

Yes . Yes bleaching trays are safe. There is no adverse effects of bleaching as long as you dont overdo it. The tooth enamel does weaken slightly while bleaching- thats why bleach supplied by the dentist contains Fluoride and you are supposed to brush your teeth with toothpaste before and after bleaching.
Yes, but.. Allleaching for teeth works basically the same way. A bleaching chemical is used to oxidize the organic debris on your teeth which is too small for a toothbrush to remove. There are varying strengths and prices. The best bet is to start off with the lowest strength and if needed work up to the next level. All can cause tooth and gum sensitivity. Rarely can they hurt the inside of the tooth.