They don't appear to be 'bad' strings (made by Martin). But upward trem use will put a lot more tension on the strings, so I'd try some of the more expensive string sets that are supposed to be able to take a lot more tension e.g D'addario NYXLs. Darco are probably fine for non-trem use.

Solder is fine for stopping string ends unwrapping, but it won't solve a problem like this - it's the centre core of the string that has broken and solder will have minimal effect, but will make the string less easy to bend.