What sequences are between adjacent genes?
No. The intergenic regions are not necessarily heterochromatin. Chromatin conformations are usually very long range and are not usually confined to a single gene. The spread of chromatin state can be prevented by insulators/boundary elements, which again are not a part of the transcribed region. The intergenic regions can also harbour distal regulatory elements such as enhancers and silencers; they can have microsatellites, transposons etc too. To know what all are there in a specific region, you can look at the different tracks in the UCSC genome browser.

In short, there can be functional and/or non-functional DNA in-between the transcribed regions. 

Repeat regions need not be heterochromatinized and conversely not all heterochromatic regions consist of repeats.