You can use KeePass, but you'll have to figure out a way to have your password vault available on other devices (can do it by using any cloud shares, i.e. GDrive). This way you'll be in charge of almost every aspect of your passwords. But you'll have to take care of backups and keep everything in sync.

There are libre off-line password managers. Variants of Keepass for example.

Indeed it's a bad idea to store passwords in a propietary system. Specially a cloud based one being hacked time to time, like 1password.

It's the choice between trusting one company (or if you self host, trusting yourself) to have their security all in order and properly encrypt the password vault. Using one password for every site you use means that you have to trust each of those sites equally, because if one leaks your password because they have atrocious password policies (eg. storing it in plain text), it's leaked everywhere and you need to remember every place you used it before.

Good password managers allow audits, and do at times still get hacked naturally (which isn't 100% preventable). Yet neither of these should result in passwords being leaked. Why? Because they properly secure your master password so it can't be reverse engineered to plain text, and without the master password your encrypted password vault is just a bunch of random bytes. And even in the extreme situation it did, you know to switch to a better password manager, and you have a nice big list of all the places where you need to change your password rather than trying to remember them all.

Human memory is fallible and we want the least amount of effort, because of that we usually make bad passwords. Your average site does not have their password security up to date (There's almost a 0% chance not one of your passwords can be found here). If you data is encrypted accordingly, it doesn't matter if it gets leaked in any way or stolen by some rogue employee, so long as they do not have your master password. So yes, I'd say that's a good idea.

The danger of using the same password everywhere is from leaks caused by poor security in one of those sites.

Passwords getting leaked are almost always unrelated to how strong the passwords are and has more to do with how those password are stored, and what protection measures they have against unauthorized people accessing them.

No one is ever going to "crack" your single password for your password manager as long as it is a strong password, though you might write it down in your wallet and lose it in a busy station, just like some administrator of a website might forget to close outside access to their mysql database containing unencrypted plaintext passwords.

In theory, yes but if you use a good password manager and have a strong master password the encryption should be practically impossible to break. The fact that you only have to remember one password means that this password can and should be a very strong one. 20+ characters with upper and lowercase letters, numbers and symbols should take centuries to crack.

You should be safe as long as your master password isn't small, less than 15 characters. The longer the password, the better. Personally what I do is use a pass phrase to make it easily memorable, and then use it as a base to inflate security somewhat artificially.

Wrap the pass phrase around in brackets or symbols; mix lower/upper case; replace (or add to) a word in your pass phrase with one from a random other language, so instead of hello you type bonjour. Bonus points if you are able to replace even a few letters in your pass phrase with fancy diacritics, or fuck it add an emoji or two.

Then again there are a LOT of other factors which go into security. Theoretically the lyrics of song are decent as a pass phrase but there's not much point if everyone knows what your favourite song is, or if you are learning Spanish then you'll replace the English words with Spanish.

Unless you're in a position where you're targeted by nations or are working extremely high profile jobs like CEO or digital security you should be safe really with all these but as I said there's a lot to keep in mind.

Just don't use your master password anywhere else than your password manager.

If your password manager only works offline, then it is impossible to leak on the internet.

Depends if you trust your password manager site more than either site you put the same pw into

This is not necessarily true.

For example, consider the case of a 1Password vault falling into the hands of an attacker. They do not have the option to just crack your password, as the password is mixed with a randomly generated value to ultimately derive the key. They would need to simultaneously brute force your password and that random value. This should almost be impossible. However, given access to a client that already has knowledge of the secret value, it would fall back to brute forcing the password.

You can go into your vault and choose a password to see if it's been exposed on the web. It's a little check mark by the password.