Depending on what tech tree your custom race is using, some weapons may be missing. There will be changes in other areas of the tech tree as well. From experience I know the Thalan tech tree is missing all the weapons you listed.

If you have tech trading on, you may be able to buy some of the missing techs from AI players, assuming they have those techs in their tech tree and have researched them.

No, but they are picked at about the same stage of setup. On the race customization page there are several tabs, including the one where you selected the terran ship designs. Right next to that is the tab where you pick starting techs; on that page, about the center of the screen, there is a line to select tech tree. This is apparently set to be the Minor Race tree by default, which would explain why you didn't have any race-specific tech, although there should be three levels of pulse cannons between the end of rail guns and the beginning of graviton drivers.

Also, IIRC a few of the midlevel weapons techs were removed entirely. I can't remember every seeing HD spike drivers in any TA game. These were mostly replaced with race-specific low level weapons, such as the Iconian scatter blaster.