In Deathwatch I notice there are rank tracks for general space marine, deathwatch, and specialty. When you are rank 1 are you eligible for buying abilities from rank one of all three types?
Yes. From the sidebar "Chapter Advances and Deathwatch Advances", page 57:
"Deathwatch-specific Advances can be selected by any Deathwatch Space Marine of the appropriate Rank, and Chapter-specific Advances can be selected by any Space Marine from that Chapter of the appropriate Rank."
As an advancing Character on Rank 1 (for example after the first mission) you can chose from all the Advancements you have stated. From the general Space Marine Advances, your given Speciality Advances and the Deathwatch Advances.
There is one exception though. Upon Character Creation you may not use the Deathwatch specific Advances, as these 1000xp you are spending represent your training before you came to the Deathwatch.
See page 27 'Starting Experience'.