Blockchain.com offers a wide range of APIs that covers most of the basic use cases:
https://www.blockchain.com/apiYou could also look into Electrum's API:
http://docs.electrum.org/en/latest/protocol.htmlIn this case you'd still need to install Electrum on one of your servers and secure the wallet yourself, but you wouldn't need to run a full node.
I think for the more funky stuff you'll probably have to look into Electrum's API rather than Blockchain's though.