1. Bitcoin core and electrum are in fact host storage wallets, so it is kind of strange that people refer to them as the best wallets.
What do you mean by "host storage" wallets? With both Core and Electrum, you generate your own private keys and you are the only one who can access them. This is generally referred to as "self custody" or "non-custodial", to differentiate from web wallets or exchanges where a third party holds your private keys for you.
Both Core and Electrum can be used as a simple hot wallet on an online device, which is the least secure way to use them. Similarly, both can also be used as a cold wallet on a permanently airgapped device, which is a much more secure way to use them. And if you are using them on an airgapped machine, you can also use them to generate key pairs or seed phrases that you then print out or write down to create paper wallets.
2. I know that this is private, so feel free not to answer it, but when it comes to you, where do you store the keys to your coins? Do you have a combination of hot storage and cold storage?
Yes, exactly that. I have a small amount of "daily spending" bitcoin which I carry on a mobile wallet. Insecure, but very convenient, and only ever an amount I can easily afford to lose. The vast majority of my coins are in a variety of more secure wallets, including hardware wallets, airgapped cold storage, paper wallets, and some multi-sig wallets involving a combination of these things. All my wallets are synced from my own node to minimize any privacy leaks.