When you are clicking on parts (Is it alt-click or shift-click or ctrl-click I am not sure as I just try them until it works.) the parts in the tray at the bottom will all show selected. So if you are trying to group 3 parts, they will all have the yellow braces around them. That is how you know you are doing it right. I have started just selecting them from down there instead of on the model, not that it makes much difference, it is just easier for me.
After you do have more than one selected down there you should be able to group them. Once you do group them however you cannot then save them as separate part. (Because it would be to convenient) If you wanted to save that as a part assembly you have to click the save button first. Group them after if you want. Saved parts do not always have a icon so just scroll to the bottom of your parts list and mouse over empty areas after the last part and see if you get a tool tip.
Clicking on change the rotation from hard point does in fact work. However depending on how you mounted your part and the kind of part there may not be any difference in some or all of the axises.
The ship designer needs some love but I have faith that Stardock will get it right.