1) What are the secure implementations of a complete node portfolio? Currently, I just know the heart of Bitcoin. Isn't there another one with a graphical interface?
I don't know of any others, apart from Bitcoin Core forks like Knots. See https://bitcoin.org for an overview of the different types of wallet software.
2) Can you use a Ledger device with these full node portfolios?
Yes, although it is currently a complex manual process using command line tools.
3) Many wallet applications like "Ledger Live" use their own servers to verify transactions. What is the case if you are using a full node portfolio like Bitcoin Core? Which servers do you ping during transactions?
Bitcoin Core is a complete node that does not rely on any trusted service. The wallet built into Bitcoin Core (or any other wallet you connect to) polls only your own node.
The node of course needs peers to download blocks and transactions, but all it needs is an honest peer. Since all data is verified locally, it doesn't matter if your peers are trying to mislead you.
4) Can the Bitcoin Core wallet censor your transactions in any way, hide or simulate them, extend them?
If the software you use yourself is malicious, it can do anything, including censoring your transactions or stealing your coins. Bitcoin Core is open source, and many eyes are looking at all the changes, but you need to make sure you're using the official Bitcoin Core binaries from https://bitcoincore.org (or https://bitcoin.org), and verify signatures (see instructions on these sites for doing so), to prevent someone from giving you a modified version.
5) In the core bitcoin, how is the mechanism for recovering funds? Is it "Hierarchical Deterministic", can you still restore your funds with a 24-word mnemonic (also by other HD wallets)?
It uses BIP32 bypass internally, so keys and addresses are deterministic. There is no starting sentence; you must make a backup of its wallet.dat file.
6) If someone, for example, has access to your computer where your main Bitcoin wallet is installed, could that person freely access and trade with it? What security mechanisms are in place for this scenario?
You can protect a Bitcoin Core wallet with an encryption passphrase. No one can send money without having access to the passphrase. Note that there is no way to recover: if you lose the passphrase, your pieces are irretrievably lost.
7) How can you restore funds with Bitcoin Core when you lost your computer where the wallet was installed? What are the different recovery options?
Is the Bitcoin basic wallet compatible with this recovery mechanism listed in github: https://github.com/iancoleman/bip39
You restore the wallet.dat file from the backup. No other options are supported.