Without installing any third-party software which I don't completely trust it functions well
Bitcoin Core is no more official than any other Bitcoin wallet software. It is just as "third-party" as anything else. It has a fancy name, and it is popular so it's currently considered to be the "reference client", but if some other full node software were to become more popular than Bitcoin Core, then, due to how the bitcoin protocol works, that other software would be the "reference client".
Is it possible to use the send toaddress and get new address functions on my PC only if I have the wallet.dat file?
To send a transaction, the wallet software needs to know what bitcoins you have. Bitcoin Core determines this by downloading the blockchain and noticing EVERY transaction that sends bitcoins to a transaction output that it knows how to access.
If you only want to store private keys, and generate receiving addresses, you can do that with Bitcoin Core without downloading the blockchain. If you want to be able to tell when you've received a confirmed transaction and/or you want to be able to send any transactions, you'll need to have processed most of the blocks (enough of them to have gotten caught up to where the bitcoins are confirmed).
I'm not interested in downloading the entire blockchain files.
Then you're not interested in running a full node. Why are you using Bitcoin Core?
Can it be done?
Sure. You can use a SPV (Simplified Payment Verification) wallet.
How do ledger wallet keys operate?
The blockchain is not replicated in this little key. And it functions!
Correct. The ledger device stores only the keys and the necessary software to communicate with other wallet software.
To use the ledger device you have to connect it to wallet software (such as Electrum). Most of the software set up to work with the Ledger are SPV wallets. The wallet software handles verifying when payments are received/confirmed as well as building the transactions to send bitcoins. When you want to send bitcoins, you first use the wallet software on some other device (such as a computer) to build the transaction. Then you send the unsigned transaction over to the Ledger device. From the Ledger device, you can sign the transaction. The signed transaction is then sent back to the wallet software which then broadcasts it to the network.