1. Rooting itself does not erase user data, but unlocking the bootloader, which is a first step in most rooting processes, will erase everything.
Some older devices would have device-specific hacks for rooting that might not erase stuff, but as OS security has developed this is less common.
I don't know your device (and Android 6 is the only major version in the last 13 years I've not used), so don't know what options or tools there will be for it. But it's likely you'll have to unlock the bootloader, and hence likely it will erase the lot.
You should always back up everything important before messing with the phone's software anyway.
2. This is Catch 22: to back these data up properly you need root!
Back up everything you can with whatever tools are available (e.g. separate backups for SMS, other messaging apps may have their own backups, there are various ways of backing up contacts).
On an old Android version like that you might find the Helium backup app works OK (I used that with Android 5, and while it can't back up everything it got most stuff). This is based on the "adb backup" command, so you could look into that as well (if you are thinking of rooting then you need to learn about adb anyway).
You can try cloud backups, e.g. Google offer backups of app data. Personally I've never had any luck with that, but you can try. Cloud backups of contacts, calendars, photos etc are straightforward.
Some manufacturers might have backup tools of their own (to back stuff up to a computer), though your phone doesn't sound like it's likely to have that.
Backups have always been a weak point in Android. Probably because it's Google, who think that everything should be in their cloud and don't really care much if you want to do differently.
3. Probably not, but I've never looked into this.
Bear in mind though that if what you are asking here were straightforward it would provide a route for a thief to access confidential information, such as banking details. So it's not obvious that they'd want to make it easy.
4. The only way to do an image backup (which is actually a set of images) is to replace the system recovery module with a custom on (e.g. TWRP) and use that to make a backup.
Installing TWRP requires the bootloader is unlocked, so you are back to the problem that you'll erase the data before you can install it.
Also, and this is extremely important, recovery modules are specific to the device model. So you'll need a custom recovery that was built for your particular model of phone. Not just another phone that has the same processor, or the same Android version, but the exact model. If you attempt to install a recovery built for a different phone the best case scenario is that it doesn't work at all: the worst case is that either installing it or using it corrupts other partitions in the phone, leading to permanent data loss and possibly making the phone unusable and unfixable (e.g. if you corrupt the bootloader that's game over).
5. If you have a custom recovery and a backup made from it that's easy: boot into recovery mode and tell it to restore the backup. A recovery backup (also known as a "nandroid" backup) backs up the ROM (system and kernel) as well as user apps and data, so this will restore the phone to the state it was in when the backup was taken.
For other types of backup discussed above, it depends on what tool or method was used. Usually this will involve using the same tool to restore. As for Google's app data backups someone else may advise, I've no useful experience of that to offer.
