Hi
" The problem now is that after moving them all out of the way, inserting the new clip at beginning - how do I move all the clips back together to seam up with the new clip? "
Grabbing all the clips or grouping them you can move them to the right and they will stay put if you press Ctrl +Alt. Now having moved all your clips along to make enough space for inserting a new one you will have a gap. Simply right click on the gap and select Close Gap in Track from the menu that comes up.All the clips to the right of the gap will shift left.
However from your original post you say you wanted to add a clip to the begining of your timeline. To do this without all the grouping phaff, left click (select) the clip in the clip bin and under the Clip Preview window you will see a Place button.Simply click the down arrow to the right of this button and select Place on Sequence at Start. (Ctrl+SHIFT+HOME).
..and when I want to cut a clip to splice on in to the middle of a video, all the following clips automatically jump the the left locked to the clip that was cut -
If you want to insert a clip at any point, move the red cursor line to the insert point, select your clip in the clip bin and as above, use the Place menu to Place on Sequence at Cursor option. You could edit the clip in the Clip Preview window before doing this using the START and END markers or edit in the usual way after insertion (like film) by splitting and deleting unwanted bits (followed by closing any gaps produced.)
Nat