1. How does the Bitcoin network guard against hackers seizing control of the system?
The Bitcoin network is a P2P network of nodes, each connected to a limited number of other nodes. Nodes pay attention if other nodes misbehave and after some threshold of misbehaving honest nodes isolate dishonest or misbehaving nodes from further communication.
2. How do miners fit into the Bitcoin network?
I'm not going into detail of mining pools or single miners. A miner is basically a Bitcoin node that assembles a so called block template, a block of the blockchain contains transactions and some other details which chain the blocks together, from unconfirmed transactions from its mempool and tries to find a blockheader hash for that template block that is below the required difficulty. Mining hardware slightly modifies certain details of the blockheader or the block content which modifies the blockheader again and ASIC hardware does the hashing until a blockheader is found which double SHA256 hash is below required difficulty.
3. How is fraud prevented on the Bitcoin network by transaction verification?
The transaction model of Bitcoin is based on UTXO, unspend transaction outputs. You can only spend such an UTXO. Once it is spend, it's gone. UTXO are inputs to a transaction and are consumed into outputs in a transaction. Those outputs are then new UTXOs. Every node checks that a transmitted transaction has valid UTXO(s) as inputs and will otherwise reject if an input is no valid UTXO.
Same is done if a miner publishes a new block. This new block is checked for validity and if every transaction in it is correct and valid otherwise the new block is rejected and not further propagated through the network of nodes.
4. What distinguishes a private key from a bitcoin address?
A private key is a 256-bit random number within certain bounds of the 256-bit space. By elliptic curve magic you derive from this private key an unique public key (details don't matter now, only that this elliptic curve transformation is a one-way function; it is computationally unfeasible to reverse this process). A public address is a recipe how to construct a spending script. In most cases it is simplified a hash of the public key with RIPMD160 hash function, this is also a one-way function, easy in one direction, computationally unfeasible in the reverse.
See here for a simplified description:
https://learnmeabitcoin.com/beginners/keys_addresses