What was the earth like in the carboniferous period, such as the amphibian, cockroach,and
Plant life?

This answer will give you some idea about the carboniferous period, but you best bet is your local library.

Aleoenvironment

The climate of the Lower Carboniferous had evolved into hot, desert-like conditions, perhaps similar to the area around the present day Dead Sea. In the later part of the Lower Carboniferous (315-310 mya) the climate changed from dry to very wet, as the sea flooded some low lying areas and evaporated into the hot air.

Plants became important ground cover in this lush new environment. The first animals with backbones (amphibians and reptiles) ventured onto land, leaving their trackways for us to find.