Boys ARE different, they have way more energy and don't CARE what you think of them. You need to find something, anything, that he likes. He's 15 months, so saying he is deliberately trashing things or destroying carpets etc is a bit much. Cover the carpet in a cheap shower curtain and let him have at it. I used this religiously at meal times, only way to stop the carpet being destroyed.
Have you spoken to your daughter about how she manages his behaviour?- if it's normal at home, and she isn't raising these issues and trying to change the behaviour, you are never going to succeed in changing this on a part time basis.
It's also been a while since you had very young children, your life and abilities are different. Accept that you possibly aren't fit enough to keep up with him and this is potentially why he's getting a hold of things he shouldn't. Maybe you need to be honest with his parents and say that you aren't up to it anymore? It's also deathly boring for a toddler to sit in all day if you aren't fit enough to go out with him (which I accept is another horrific challenge in and of itself), and maybe he needs a toddler group/ nursery so he can mix with other children and model (hopefuly) better behaviour?
Boys need activity and new things to hold their interest - simple and cheap stuff works - make your own playdoh, lets him mix things, add food colouring etc, then gives him a toy to play with at the end of it. Making a music maker from pasta or rice in an empty container.
Chalk for the path - let him scribble away to his hearts content and then you can wash it away when he goes home. Boys like mess and play. If you aren't fit to do the running around stuff, then you have to do the messy stuff I'm afraid. Just prepare for him coming to minimise the mess, but let him do fun messy stuff that he can enjoy and isn't going to make you cross. He'll see that gran isn't always complaining, and that you can enjoy one another's company.