Hi LingDiLong welcome to the wonderful world of guinea-pigs! They are fabulous.I got back into guinea ownership after having them as a child.My DD wanted a pet (she fancied a hamster but I talked her into my way of thinking and last year we got 2 one-year old brothers from a rescue)
WRT to run- mesh won't protect them.It's got to be cat and fox proof. We have a 7' rabbit run that is chrome coated metal (it can fold up and has long 'pegs' to secure it to the lawn)They won't dig but you need to make sure nothing digs its way in.
This morning I found the foxes had shredded the newspaper and tried to pull the blanket through the bars (my hogs were safely in their Pighouse) so don't underestimate the damage they can do.
I don't have a hutch for my pigs.I looked at hutches, all the advice was put them in a shed or garage for winter. So I gave our pigs the DC old wooden playhouse. It's got a floorspace of 5'5" x 5'7". My DH made a few alterations to it. They have a big haybox with a wooden lid (made out of an old bookcase). A wide wooden ramp leading to the lid (both covered in old carpet). He made a half wood,half screen door (old fireguard) to fit inside when the wooden door is open.And one of the windows can lift out (mesh and wire fencing lined to protect from foxes/insects).
There's a barrier across the doorway to stop any chance of escape.
The house itself is slightly raised on a wooden plinth (in the rain they were fine.The garden was like a river)
There's lots of website that give you cage size recommendation.Most say 4'x2' minimum. (I have an indoor cage for winter that is 4'x2' and it's barely adequate. It's just for winter night)
They like lots of hiding spaces. They need company but they need there own space (especially if you decide on 2 boys). If they get in each others faces,that's when fights start.
And they are untidy little ratbags. Unlike rabbits or other furries. I had one GP who would pooh on a flat newspaper lined tray.Most just pooh where they walk.
The Rescue you are looking at will give you an idea of what you need space wise. Though some of them insist you keep them indoors.
C&C cages (the ones you build with the square grid sides) can adapt to the size and shape you want, you can do different levels, and you can add space. Some cages have upstairs but if the ramp is too steep they won't attempt to climb so might be stuck on one level.And the sleeping quarters are small.DD and I were looking at cages today.Just out of noseyness
You need to allow space for food bowl, waterbottle/bowl, veg plate, hay
We find the indoor/outdoor works for ours. They have a morning hello and breakfast.If I'm home they go out in their run.Otherwise they might get a couple of hours at evening. A cuddle at night .
In winter they go out in the morning to the Pighouse with heating (hot waterbottles in fleece covers) in loads of hay.
And if you do put them in the shed, try and have a light.Makes a huge difference!
Good Luck.
Lots of guinea-pig people on here will be along.
Some keep indoor. Some outdoor. Some both like mine. It all works. Alot depends on the age of your pigs too. Very baby babies would be need to be protected more than adults, but summer is a nice time to get them..