1) Which secure full node wallet implementations exist? Currently I just know of bitcoin core. Is there not any other one with a GUI?
I don't know of any others, apart from forks of Bitcoin Core like Knots. See https://bitcoin.org for an overview of various types of wallet software.
2) Can you use a Ledger device with those full node wallets?
Yes, though it is a complex manual process using command-line tools for now.
3) Many wallet apps like "Ledger Live" use their own servers for transaction verification. What is the case if you use a full node wallet like Bitcoin Core? Whose servers are you pinging there during transactions?
Bitcoin Core is a full node that does not rely on any trusted services. The built-in wallet in Bitcoin Core (or any other wallet you'd connect to it) is solely querying your own node.
The node of course needs peers to download blocks and transactions from, but all it needs is one honest peer. As all data is locally verified it does not matter if your peers are trying to mislead you.
4) Can the bitcoin core wallet censor in any way your transactions, hide them or simulate, prolong them?
If the software you yourself use is malicious it can do anything, including censoring your transactions or stealing your coins. Bitcoin Core is open source, and many eyes review all changes in it, but you have to make sure to use the official Bitcoin Core binaries from https://bitcoincore.org (or https://bitcoin.org), and verify the signatures (see instructions on those sites for doing so), to prevent someone from giving you a modified version.
5) In bitcoin core, how is the funds recovery mechanism? Is it "Hierarchical Deterministic", can you you always restore your funds with a 24word mnemonic (also by other HD wallets)?
It uses BIP32 derivation internally, so keys and addresses are deterministic. There is no seed phrase; you need to make a backup of its wallet.dat file.
6) If someone for example gets access to your computer where your bitcoin core wallet is installed on, would this person be able to freely access the wallet and transact with it? What security mechanisms are in place for that scenario?
You can protect a Bitcoin Core wallet with an encryption passphrase. Nobody can send funds away without access to the passphrase. Note that there is no way to recover: if you lose the passphrase your coins are irrecoverably lost.
7) How can you restore funds with bitcoin core when you lost your computer where the wallet was installed on? Which different recovery options exist?
Is bitcoin core wallet compatible with this recovery mechanism listed in github: https://github.com/iancoleman/bip39
You restore the wallet.dat file from backup. No other options are supported.
