1. What kind of care would they need when we go out of town?
2. If my 11-year old wants to hang out in the aviary, how will the finches respond. (Of course she'd love for them to alight on her finger.)
To answer your questions:
1. When out of town, so long as the feeders are full and you have multiple water dishes they will be fine. Of course you will need to have someone come and change out the water daily as contaminated water can lead to several health issues. If you have bird water bottles you change those less often but I dont know if finches would take to those.
2. When building the aviary you will need a safety door or else you risk the birds flying out and escaping(no matter how careful you may think you're being). You will have to enter the aviary for feeding and watering the birds so they will get accustomed to having you in the aviary. As of having them sitting on fingers you will need hand tame birds which are hard to find in finches. Or you can go through the sometimes very long process of taming them.
Is your aviary going to be heated? I don't know how the weather gets in Texas during the winter but you'll definitely have to cover the aviary with plastic to keep it warm on the inside as well as prevent drafts and the like. If it gets too cold you will have to invest in a heated aviary.
Also you must predator proof the aviary, finches make tasty snacks for most land and airborne predators. You should use mesh wire that's about 1/2'' to 5/8''.
An aviary is a wonderful idea but it is a lot of work to maintain.
How big do you plan on making the aviary?
And how many finches are you going to house in it?
I should also note that finches breed like crazy. You'll definitely have double the amount of birds in a year so you have to consider what you will do if the population becomes to high(you can find homes for them).
I have two aviaries. 1 is 12x8 and one is 8 square. They house cockatiels and English budgies. I live in Florida and it can still get cold here and breezy but thankfully I do not have deal with snow. We use a heavy duty tarp which we modified to fit exactly and hang it from screw hooks when needed. We can drop it easily during the day times when it's warm and quickly put it back up in the evenings or on breezy cool days. We can also roll the tarps in spring and store them under the house until the next winter.
Although we live in florida we do get cold on rare occasions. We use oil filled radiator heaters since they are a far lower fire risk than heaters with elements and there's no where the bird can really get caught in the heater. Remember heat does rise, so having a floor heater still gets the top of the aviaries warm.
We used 1/2 inch hardware cloth on the aviaries. it's been great about keeping out predators but small snakes could still get in. I unfortunately left the door open one night when we were in there late and only discovered 3 days later when cockatiels were missing 3 red rat snakes had taken up residence. It sucked. But when having anything outdoors you will get all sorts of predators and pests. I never seem to have a mouse or rat problem. It may be because I use button quail on the aviary bottoms to help keep the seed cleaned up. I have heard of rats killing buttons but since a large rat can't get through 1/2" mesh I haven't experienced that.
I can only speak from experience on how my cockatiels and budgies act when I go in the aviaries. While a few of my tiels are tame most are not and none of the budgies are. The tiels have not got a problem with me going in and out and they have absolutely no desire to leave, even with the door open. The budgies are more skittish and will flutter around, but again they do not want to leave their flock and even if one does get out (and it's happened) they come right back and want back inside.
Wintertime this year, and summer 2 years ago when finishing my small-bird aviary. My chicken/turkey big-bird coops are the same design but a lot bigger. I've kept and bred canaries in an unheated attached garage. They are fine as long and they get adjusted to the lowering temps slowly as winter comes on (don't get moved from heated house to unheated garage overnight) and there are NO drafts. They were fine at 40F. If you hand raise a canary chick it will land on your finger the rest of it's life in my experience.