Sounds like you have two problems to solve.
For your DNS I agree dnyDNS is a great solution for what you're looking for provided your address doesn't change often. Before going down that road I would find out what the lease time your service provider's DHCP server is handing you for your address. Many keep those short because they have a limited pool of addresses to play with and they cost money if they are not cycling off freed up addresses quickly meaning that address you're issued is bound to be released and handed back out should your system go off line.
If it's that important to be able to access your system and not have the IP change on you I'd really consider getting a static one issued.
Now as for your power, have you considered an IP based PDU? Basically it's a power strip you can control over wire, and power on any outlet remotely. Super handy to have. Unless you blow a breaker you'll always be able to bring your system up.
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