The link shows stairs.

Yes, but with a tree on the side. I wonder if those "containers" where the plants be will need some sort of draining...

Consult a local botanist. Seeking inspiration from images from Brazil or Portugal may be inappropriate for your area.

But in terms of construction and materials would it be fine to have some this? or would lead to long term problems?

I think you need to explain yourself. There are lots of ways to bring greenery into your home without it being part of the construction. If I was designing a new house and this was a big item for me I would design a house with an atrium. We had one and if I was to build a new house I would do the exact same thing. You can do it open to the sky or covered depending on where you live .For that it then can become part of your living space . We used ours for outdoor dining .This one is quite interesting. As for drainage , yes you need to design for that

Yes, an Atrium.

It can be outdoors with the house organized around it, or interior. Outdoors is more trouble free.

An impervious floor and drain are required, as well as sufficient natural light for plant growth.

Choose plants that perform in lower lighting conditions and which are managable in size.

Good luck.

You could do an atrium, as suggested and shown above, or you could go a little larger and do an interior courtyard:

I have a live wall installed on a porch on the exterior of my house. The website is livewall.com and it might give you the look you are looking for. They are kind of pricey, but they come with essentially everything you need in one custom unit. The indoor version has extra drainage provisions (that I have no experience with). My semi-outdoor version has drainage, but I don't believe it is as robust as the indoor version.

The pics on their website are better than the pics I have as my plants were just installed last week and aren't fully grown in.

I did buy an extra set of the containers so the nursery can pre-grow my next set up while we are enjoying what is in there now. I am thinking I will rotate it quarterly as the seasons change...

Eichler plan with exterior atrium:

When designing something like this, in addition to light requirements and drainage, also consider invasive roots (some trees' root systems will plow through a foundation, some won't), root range (some plants will die early if they don't have enough space to branch out; some plants like being a little more root bound) and leaf/flower litter. Pest attraction can be an issue, as well as disease (some plants are more prone to things like aphid infestation or black spot). These aren't usually devastating, but fragile plants can increase maintenance, which may or may not be your cup of tea.

I love atrium houses. Absolutely love 'em. I don't want to discourage anyone from building one. None of the things I mentioned would deter me from building an atrium (or large outdoor courtyard surrounded by rooms), but I wanted to mention them to give you more to mull over. :-D

Be sure you think about resale value... may hurt (or help) resale....

I have friends that have a modern house with an actual open space to the ground in the living room slab, and there is a large tree growing there. It looks kind of cool, but the tree has grown up to the double height ceiling so it needs to be trimmed, and part of the tree looks like it might be dying. If they have to remove it, it will be quite a project. I also wasn't sure if I would want a space in my house that burrowing critters could come up through.

I did tour another house when I was looking to buy, and it had a central atrium that was glass on all sides with access through a glass door. It was natural dirt, so you didn't have to worry about drainage. It allowed you to see what was growing from various locations in the house without having to be concerned about animals coming in to your house.

Planter boxes of concrete which are integral to the inside of a house look cool, but I think the care of the plants, the drainage, and other complications can be a continuing demand.

If you choose an atrium open to the interior spaces in the home, be sure to carefully consider how moisture will be managed.

I love the idea of an atrium in the middle of a home, but I would definitely be concerned about water control during storms. I'm sure it can be done for a price.