How likely would it be that I could pull off a shotgun shot to the head and a handgun shot to the heart at the same time without flinching and Fucking the whole thing up?
Very unlikely. That's a lot of shooting to do all at the same time. And it's a pointless exercise: one correctly placed shot is all you need.
And where can I go to learn to shoot these things,
That will depend on where you live. If there's a local gun store, they'll be the ones to ask. Or you might check the NRA website? I don't endorse that organization, but they would likely have info.
how many questions are asked at ranges,
Aside from "What do you want to learn?", not many. Depending on the range, they want to know what sort of gun you're shooting, so you don't punch holes in the backstop, and they might express curiosity about your gun because it's interesting to know what other people are shooting, but privacy is a pretty important thing for us gun nuts, so you're not likely to get quizzed about motives or motivations.
do I need to bring my own guns, (don't want to do that),
It depends on the range. Some rent guns, sell ammo, offer tutorial, and have a gunsmith on hand; others are bare-bones BYOG and make sure you pick up your own spent casings.
how difficult is it to purchase ammunition?
Not at all, if you're over 21. Under 21, you might be able to buy online with appropriate duplicity.
Will I go through the same background questioning as I did when I purchased the actual firearms?
For ammo? Not even a little.
If you have already purchased the firearm, you're in the clear. That's the hard part. If you have purchased a gun, anyone interacting with you takes for granted that you've officially been vetted and there's no need for further questions.
What are the odds that these words I'm typing will set off some kind of follow and I'll be flagged or something?
Odds are pretty much zero.
How do I do this with the least mess?
The best solution I've come up with is either in the bath/shower, where the mess (inevitable with gunshot) can easily be rinsed down the drain, or outside.
I want the comfort of my home, but I'd rather live forever than have my family find me or have to clean things up. But I do want to be found.
I hear you. I have the same conflicts. The best scenario I've worked out is to shoot myself out in the yard, someplace where the mess can be unobtrusively hosed into the ground. I live alone, so a delayed delivery email will notify the sheriff to come tend to my body. For me the trick will be getting found by the sheriff before my body is found by the coyotes.
Alternately, there's a spot in a local park, out of sight of the populated areas but not out of earshot, where I might shoot myself. No one needs see the event, but the sound of the gunshot should get people's attention and I'm confident the police will arrive quickly. At worst, it will be a stranger who catches sight of the body, and I'm confident the police will be on their way soon after that.