Your new employer has to file a H1B petition de novo, with the only difference that you are not subject to the lottery any more. USCIS may well take the view that your new job, even if it is doing the same thing, is not a specialty occupation any more.

SWE->SWE is relatively safe. And there are a number of significant differences between a truly "de novo" petition and a transfer. While it is true that you need a sponsor, and therefore a petition needs to be filed, you don't need to wait for it to be approved to start working at the new place. You can also have multiple prospective employers file petitions, so if one gets denied another might get approved, or if you get a better offer you get the option of picking and choosing. The process is still fraught with uncertainty (particularly if H1-B worker brought their wife and kids along), but you aren't really "starting from scratch" or undergoing the same degree of scrutiny.

It used to be worse, BTW: it used to be that you couldn't really change jobs at all, and until fairly recently, your spouse (who is typically on H4) couldn't work. Or to be exact, she/he could, but couldn't get paid (volunteering was OK).

It isn't a "notification", you still have to file for a new visa regardless of how similar the new position is. And it can very well be denied. The only concession is that you won't be subject to the annual limit and won't have to go through the lottery.

It's a one-time grace period, not 60 days cumulative over a lifetime. The second time you get laid off without a new job in hand no grace period for you.