Part 1 -> How do I fix this?
Try clicking on the two items you want to mate and use the "Aligned / Anti-Aligned" buttons in the mate dialog box (at the bottom of the "Standard Mates" section
Part 2 -> Move with Triad
I tend to use this more for positioning components "close" to one another rather than trying to match up surfaces. Note that it's just as easy to " left click and hold" to position components as well.
Part 3 -> Missing Triad
If you right click on the component and look at the pop up menu there will be a set of down arrows at the bottom of this menu. If you click on that does the triad appear back in the menu?
sounds to me like you want to rotate in fixed angle increments.
You can right click on a triad ring and type 180 to turn the object. (90 for a quarter turn)
You can also investigate the Show Rotate Delta XYZ interface in the help file
you should find moving/rotating objects roughly close to what you want and the align/anti-align setting setting should give you what you need. later because many mates have more than one mathematical possibility, you will need to learn how to design/build models with reliable sets of mates.