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LSF is not a distro. It is a instruction manual and teaching aid. Don’t use it as a base for you main OS. And IMO Gentoo does not really teach you more then Arch does. It gives a bit of flexibility that not many care about (how things are compiled) at a very big cost (of having to compile everything yourself). I would not use either unless compiling things is your hobby. Sure, try them in a VM if you really want to. But I would not really consider that moving on from your current distro nor do you really need to do that.

You are not considered to be working somewhere until you have signed a contract and after the start date on that contract. Accepting a offer is not signing a contract. You are not working at the new place yet. You have no obligations to do anything at that point. You just need to have stopped working at your current employment before your start date. You definitely do not need to quit before accepting the offer. No where I have worked requires that.

You are right. You cannot onboard a new job before you leave your old one. Accepting an offer is not part of the onboarding process though. It happens before. After an interview process the company makes an offer. The candidate can then accept or reject it. But that is really all informal. You can then negotiate with them for an official start date and contract. You just need to ensure you can hand in your notice and work the rest of your notice period before the start date of your new contract. I don’t know anyone that would hand in their notice before accepting the initial offer of a company. At least here in the UK.

You assume they don’t already have a job and we’re just looking for other opportunities. Not everyone is unemployed before they apply for other jobs. If anything that is a good time to look as it gives you stronger position to negotiate from.