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Lets assume that the metal sheets are about 1,000km wide and that they extend about 500m up and about 300m down. Lets also assume that any problems with lighting have been solved in this case so the ground between the metal slabs are about as well lit as the ground outside the metal sheets.

If the seeds of trees that normally grow to 30m high were planted between the metal slabs would they be able to grow into trees considering that in one of the dimensions they only have 1mm of space? Would it be possible to put insect eggs between the metal slabs and have them grow into insects? Assuming we could grow mammals including humans in test tubes could humans and other mammals be grown in test tubes that only had 1mm of space to grow in one of the three dimensions provided they had plenty of space to grow in the other two and then be placed in between the two slabs and raised by robots?

What kind of effect would growing between the two slabs have on development? How would a culture living between the two slabs develop? What effect would the two slabs have on evolution for the animals and plants having to live between them?

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    $\begingroup$ 1mm wide and 800m tall would have a very serious problem with access to daylight. "Plants", including trees, require light for photosynthesis. This puts serious constraints on anything resembling plantlife as we are used to it. Algae might be able to make it (though I still doubt it), but trees would definitely be stretching it... $\endgroup$ – a CVn Oct 11 '15 at 14:07
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There are a lot of questions in your question.

Is life possible ?

We already know life is possible as micro organism in such a tight place.

I think it's possible for plants and insects.

In our environment, lichens and moss have no problem growing into very narrow spaces. Some leaf plants have very small roots growing between rocks, or between the cracks of the walls and roads. Once they reach the light, the leaves grow, and most of them are flat. If they have enough nutrients and enough light, they can grow. You assume there is enough light, but I'm concerned with the ground and nutrients the plants will need. When the environment is too tight, earthly plants do not grow big. They can live, but they stay small. They are designed by evolution to find what they need in a 3D world and don't prosper when they don't have enough space.

For insects you can grow an ants colony in a flat design like that : Flat ants.

I don't know whether ants are happy about it, thought. But they can live inside so, as long as your ants/insects are small enough, they can live and reproduce in your flat world.

Big plants and mammals ?

A big plant like a tree has big roots. I'm not sure whether a big tree could grow, but very tight and very long roots could bring a "big" plant the nutrients it needs. I'm not biologist, but I think a tree in your world could look like a big round surrounded by roots/branches with both leaves to gather energy from the light and roots to gather nutrients from the ground. As they die because there is no more nutrients, leaves and roots would decompose and become new nutrients for smaller plants (or even the "tree" itself), like in our world.

But I'm not sure for bigger animals. Mammals ? How could they find the place to reproduce, to give birth ? Either the animals are small enough to lay tiny eggs, or grow on themselves like corral, or duplicate like cells, but a very complex creature have complex organs that take a lot of space. Thus, deprived from the 3rd dimension, they should be very large to have the same organs. And they need energy and food, so now it's up to your imagination : are they going to eat plants ? To hunt ? How ? How could even eyes or ears work ? Or hands, tooths, all the ways to interact with environment ?

Society ?

I think your question is very interesting but become to broad when coming to society. It's too different from our world, so here are a few thoughts but not a complete answer :

What do they need to live ?

If they are big enough to have a conscious, human-like brain, they will need space and living with each other will take space.

Moving a big organism seems difficult in this environment as plants have roots everywhere, you can't just step over it, you have to go through and break it, damaging your potential best source of food every time you step in "the woods". Living in community does not look like a good living strategy as this many "people" moving around will destroy the environment very fast.

How could they share information ? They can't draw, write... So either only communicate by touch/sound, thus their society could not evolve very much. Or they find a way (might be by sculpting edges of the flat-trees, like binary digits ?) to write something down. But it would not be as advanced as us with that.

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