There are two different types of accounts in Ethereum: externally owned accounts (EOAs) and contract accounts. EOAs are controlled by users, often via software such as a wallet application that is external to the Ethereum platform. In contrast, contract accounts are controlled by program code (also commonly referred to as "smart contracts") that is executed by the Ethereum Virtual Machine. In short, EOAs are simple accounts without any associated code or data storage, whereas contract accounts have both associated code and data storage. EOAs are controlled by transactions created and cryptographically signed with a private key in the real world external to and independent of the protocol, whereas contract accounts do not have private keys and so "control themselves" in the predetermined way prescribed by their smart contract code. Both types of accounts are identified by an Ethereum address.
Smart contracts are typically written in a high-level language, such as Solidity. But in order to run, they must be compiled to the low-level bytecode that runs in the EVM. Once compiled, they are deployed on the Ethereum platform using a special contract creation transaction, which is identified as such by being sent to the special contract creation address, namely 0x0. Each contract is identified by an Ethereum address, which is derived from the contract creation transaction as a function of the originating account and nonce. The Ethereum address of a contract can be used in a transaction as the recipient, sending funds to the contract or calling one of the contract's functions. Note that, unlike with EOAs, there are no keys associated with an account created for a new smart contract. As the contract creator, you don't get any special privileges at the protocol level. You certainly don't receive the private key for the contract account, which in fact does not exist—we can say that the smart contract accounts own themselves.
Importantly, contracts only run if they are called by a transaction. All smart contracts in Ethereum are executed, ultimately, because of a transaction initiated from an EOA. A contract can call a contract that can call another contract, and so on, but the first contract in such a chain of execution will always have been called by a transaction from an EOA. Contracts never run "on their own" or "in the background". Contracts effectively lie dormant until a transaction triggers execution, either directly or indirectly as part of a chain of contract calls. Also, smart contracts are not executed "in parallel" in any sense.
Transactions are atomic, regardless of how many contracts they call or what those contracts do when called. Transactions execute in their entirety, with any changes in the global state (contracts, accounts, etc.) recorded only if all execution terminates successfully. Successful termination means that the program executed without an error and reached the end of execution. If execution fails due to an error, all of its effects (changes in state) are "rolled back" as if the transaction never ran. A failed transaction is still recorded as having been attempted, and the ether spent on gas for the execution is deducted from the originating account, but it otherwise has no effects on contract or account state.
A contract's code cannot be changed. However, a contract can be "deleted", removing the code and its internal state (storage) from its address, leaving a blank account. Any transactions sent to that account address after the contract has been deleted do not result in any code execution, because there is no longer any code there to execute. To delete a contract, you execute an EVM opcode called SELFDESTRUCT (previously called SUICIDE). That operation costs "negative gas", a gas refund, thereby incentivizing the release of network client resources from the deletion of stored state. Deleting a contract in this way does not remove the transaction history of the contract, since the blockchain itself is immutable. The SELFDESTRUCT capability will only be available if the contract author programmed the smart contract to have that functionality. If not, the smart contract cannot be deleted.