I made this image for making it easier to understand what I'd like to learn how to do:
You can use basic illustrator tools!
First Objective: Shift curve along line.
First desect everything and then press A to select Direct Selection Tool.
now once you have done that, just select the area which will include that 3 anchor points of your path.
now press and hold Shift and drag your mouse pointer to any one of those selected anchor point. and while holding Shift, drag those anchor points horizontally and it'll shift your curve on straight line. (Bonus: Don't press Shift if you want to move it anywhere, watch below GIF)
Second Objective: Change width and angle.
same as above, just select the anchor point which is on straight line and move them far as long gap as you want.
Select all those anchorpoint (3 in my case) and go to properties panel. there you'll find Align tab and in that align tab select "Horizontal Distribute Center". this will adjust all anchor point having same spacing between them.
now just drag the center curved anchor point and drag it up/down while holding Shift to adjust height of curve.
if you want Symmetrical then the additional steps you can use it to use guides on both side on center curved anchor point and then drag there constraints till the guide. this will keep the curve symmetrical.
What if the path given doesn't have the curve I want to move exactly 180° parallel to the ground? Is it still possible to shift the curve along the line it belongs to even in that case?
Yes it's quite possible and you'll have to use just a little more tool (i.e. lasso tool and rotate tool)
First go and play with lasso tool which can be activated by Q .it'll help you select particular area in free form mode while direct selection tool only allow square selection and also it'll select guide too which we don't want here.
Okay so first drag a guide hori/vert. Select rotate tool by R, now a crosshair will appear on guide, move that crosshair on nearest line coordination and then just rotate it to parallel(don't worry just follow along gif)
okay now do same method as first and more anchor points by selecting lasso tool; just make sure you don't hold Shift this time. and that's it. your 4th question will take time so I'll edit answer after few hours!
I also noticed that "Horizontal Distribute Center" doesn't work in my case with a soft curve, since there is no anchor point in the middle of the curve.
As I known, my first answer was only specified for single anchor point and horizontal line but I've shown Horizontal Distribution because all of the tools in that align panel comes handy many times thus at least you should be familiar with tools :D anyway now let's come to complex shapes.
in Edit 1, It seems my point of using guide is clear as you mentioned that we should rotate guides and rotating art itself for better perfection, So I won't explain more of guides here (You can see in my below gif that I followed your suggestion and used guides on anchor points)
So, In your case if I would've more that 3 point in my curve I would just remove all curved anchor points and then just create a one then centering those 3 points as I mentioned in my first answer and then just dragging curve out of it! (P.S. I didn't really horizontally centered anchors since this was just an example and you already know the rest part of it :D ). Cheers :)