Since my card is on exFAT atm can it be seamlessly converted into FAT32 without losing any data (like just convert and not format) or do I need to backup all the data somewhere else first, format the card to FAT32, and then move the backup data back to the card?
Option two. There is no such thing as "converting and not formatting" a data partition to another format (hint: it's in the name).
Also can installed games on a exFAT card be higher than the 4GB limit that FAT32 has
Yes, but on a FAT32 partition they can also...
in the sense that when a game is installed they can turn into a single file larger than 4GB
No, because an installed game is not a single...
or the installation process will always make sure to install in chunks smaller than 4GB regardless if the source file was bigger than 4GB?
Okay, do you think that installing a game is just copying a .nsp file to a folder? That's not the case. Think of the big files you install as archives, either compressed (.nsz, .xcz) or uncompressed (.nsp, .xci). Installing a game is akin to extracting that archive, which basically contains a bunch of .nca files, into your Switch's internal storage or an SD card. No .nca file is ever going to be larger than 4GB. Why? Because Nintendo officially supports using a FAT32-formatted SD card, and it's not very "officiallysupportish" of them to distribute a game that can't be installed on a FAT32-formatted SD card. Also, it doesn't make much sense from a development standpoint either. So don't worry about that.
Anyway, is there any homebrew that by nature will only work under FAT32 and not exFAT?
It's not about whether it will "work" under exFAT. It's more like playing Operation. Eventually, some software is going to make that poor patient's nose go red as Rudolph's, and though it's Nintendo's fault for shipping a shitty driver, you're the one that's going to lose his data. So it's best not to risk it.