Confusion about the brain regions - temporal cortex, hippocampus and and white matter
"Lobe" refers to a section of the cerebral cortex, which is the outer part of the cerebrum. The cerebrum contains several structures, but the one that most people care about is the cerebral cortex, which is the 2-4 mm thick layer on the surface of the cerebrum. The cerebral cortex (or "cortex") is divided into several lobes, e.g. temporal lobe, occipital lobe. So, cortex and lobe are not the same thing, as the lobes subdivisions of the cortex as a whole. (See https://en.wikipedia.org/wiki/Cerebral_cortex)
The hippocampus is a region of the cortex in the temporal lobe. Therefore, it consists of gray matter. However, it is connected to several other structures by white matter (e.g. the fornix), and these structures are related to the hippocampus. (See https://en.wikipedia.org/wiki/Hippocampus#Anatomy)
Forebrain refers to the cerebrum as well as other structures, such as the thalamus and hypothalamus. The forebrain is defined as the telencephalon (which is basically the cerebrum) and the diencephalon (which contains the thalamus and hypothalamus). (See https://en.wikipedia.org/wiki/Forebrain and https://en.wikipedia.org/wiki/Diencephalon)
I don't know what you mean by the temporal cortex having "the long side of it outside the temporal lobe", but the temporal cortex and temporal lobe are basically the same thing.