Tl;dr: Bitcoin is cash, not a bank account.
Mints don't keep track of which people hold each of their coins or how much of their cash each person holds.
Mints provide users of their coins with some ways of distinguishing genuine coins from forged coins.
How are funds verified?
They aren't. The Bitcoin network doesn't know about or care about funds. It mainly knows about transactions. From knowledge about transactions comes knowledge about "coins".
I understand that transactions go into mempool
Yes, unconfirmed transactions go into the individual mempool of each node. As answers to other questions have stated, there is no coordination of mempool contents across nodes.
and a miner "verifies" the transactions
Well, all nodes verify transactions and not just miners. It isn't one miner. You may be conflating verification with confirmation. These are two very different things.
such as ensuring the user has the appropriate funds.
No nodes care about whether users have appropriate funds, the network doesn't really know anything about users.
and correct signature.
Yes, any signatures in the transaction are verified by every node separately. The system was designed so that no-one has to rely on some stranger doing verification for them. Signatures don't really identify actual people though.
Specifically, how are available funds verified?
Each node separately keeps track of its own list of all the unspent "coins" that exist globally. By "coins" we usually mean unspent transaction outputs (UTXOs).
It seems like you would have to iterate through every relevant transaction in every block to determine whether the funds have already been spent or not.
This seems quite time consuming so surely there is a shortcut to verifying a user has available funds to spend.
No one care whether users have funds. We only care that coins are real (unspent). Its like cash. It is cash. When you buy something with cash, no one cares how much money you have, only that the cash you give them is not a forgery.
There is no real shortcut. Every nodes keeps track of a list of all genuine (unspent) coins.
lightweight nodes (SPV nodes) do rely on strangers to do some things for them - and consequently they are less secure.
How does a miner verify ...
The same way every other type of node verifies.
... that the funds are available and not already spent without iterating through the entire Blockchain?
By iterating through the entire blockchain once and creating useful, but much smaller, lists from it. Also by indexing those lists for faster retrieval. So this would include an indexed list of known good verified UTXOs that are acceptable to this node as inputs to future transactions it receives data about.