Tokens have contract addresses, see https://support.metamask.io/kb/article/24-what-is-a-token-contract-address
But is there some "base" contract that manages the sending of plain vanilla ether?
No. Ether is the basic currency, it managed by whole Ethereum network.
Almost(see comments) any interaction with contract is actually sending of ether(maybe 0) plus some data.
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Thank you. So does this mean if I start my own test network blockchain then the origin block will not have any code in it? – William Entriken Dec 23 '17 at 16:42
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Yes. Genesis block only contains some configuration data and balances of wallets. – Andrey Putilin Dec 23 '17 at 16:50
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1@AndreyPutilin A small edit to consider for technicality: to interact with
constantmethods (and nothing else) in contracts does not require a transaction and thus does not involve sending any ether (not even a 0 ether transaction). So "Any interaction with a contract that modifies the blockchain is actually ..." – lungj Dec 24 '17 at 17:13
