You can go back to the TV World at any time everyone's free/available (which is most of the time), it won't initiate a rescue, as you can tackle the floors at your own pace. All that matters is that you rescue the people in time before the fog lifts, but unlike Persona 3, it will automatically take you there when you're close to running out of time so there won't be a game over (as in P3 NPCs would die if you didn't rescue them from Tartarus in time).
I wouldn't say it's something you have to worry about (as I've managed the "Izanagi only challenge" before), but it is fun once you get your head around it and will make the game a hell of a lot easier as when you fuse Personas a number of things happen:
Your Social Link levels increase the levels of the Personas of the respective Arcanas.
On certain days (which can be seen under the Fusion Forecast option in the Velvet Room), fusions will gain bonuses like increased stats, additional skills, etc. so at times it's worth planning when you're going to go fusion crazy to get the most out of them.
The fused Persona inherits skills from the Persona used to fuse it, and while there are many, many combinations for the same Personas, you might want to plan which ones you're going to use as the skills they get can drastically change their playstyles and really make some unstoppable Personas (especially when you get ones that auto-cast every buff at the start of a fight and heal your character to full HP/SP after a battle).
It's worth noting that some Persona can only be fused once you've maxed a Social Link and received the items you get from doing so, so if you're reading threads about which Persona to use/fuse, bear in mind that some of the listed ones won't be fusable until much later.
Generally the Persona you fuse will be stronger than the ones you gain from the shuffle, even if they're the same Persona. For reasons such as this I'd recommend slowly trying Fusion every now and then to get the hang of it since you can really get out of tight situations with some inherited skills and such (like inheriting an immunity to an element that the Persona is naturally weak to to make them have no vulnerabilities that can be exploited), but one of the useful new features they added is the "Search" function in the Fusion menu where you can see all the possible combinations you can fuse with the Persona you currently have in your party as opposed to having to mess around with different "spreads".
(it's been a while since I played it last, so some of my info might a little off
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Edit: Forgot to mention - while you are able to get into Fusion from the get-go, bear in mind it does absorb a large amount of cash, so be a bit weary in-case you go overboard and don't have enough money for other spendings.