Bicycle Mechanics - Downshifting paddle on shimano brifter doesn't wind spool

I just got a new (to me) roadbike with ultegra brifters. I think that the bike was in a moderate crash, because the logo faceplate of the shifter is a little loose, and the whole assembly is rattly going over rough pavement. How do I tighten it up to lose the rattle?

Possibly related: on the stand, everything works fine. But on the road, the downshift paddle for the rear derailler does not actually pull and cable unless the brake lever is slightly nudged inward. THat is, I need to swing the paddle out while bringing it in to me slightly. Is this common? How do I fix it?

Before you ask: cable tension is fine when completely upshifted, indexing works great. I *think* that the problem is solely in the shifter, but I'm open to suggestion.

Thanks!

Older Ultegras are famous for rattling, not much can be done about it. You could try popping of the faceplate and tightening the screw underneath, it may help.
The other problem seems to be a somewhat common failure mode for these shifters. I haven't found cure, please pass it on if you do. I believe your suspicion about it being connected with a crash may be right, but have no proof except a little anecdotal evidence gleaned from googling this problem on the net.
There is a webpage out there that shows how one guy cured this problem by using a rubber band around the brake lever, I get around it by holding the brake lever with my index finger while pusing the paddle with my ring finger. Only works from the hoods. I use thumb and index finger in the drops.

Unlike women, bikes are simple and easy to figure. If it has been crashed, there will be scratches.

The rattle can probably be fixed with a bit of clear caulk or double backed foam tape, doesn't mean the bike was wrecked, this is a common problem.
As for the rear shifter, it's the brake lever that pulls cable and performs the downshift, the smaller paddle shifter releases cable and performs the upshift unless you have a Rapid Rise low normal mountain derailleur. Look for a small screw on the underside of the shifter that may be loose or missing. Also flush out the shifter with WD40, use the small tube and squirt it into the shift mechanism. Or you may need to replace the shift cables and housings, especially the short housing at the rear just before the derailleur.

Al

I had the same problem. Tried to downshift and no pull, after a couple of repetitions it worked temporarily.

The solution was to flush the brifter with WD40. I put the bike in the stand, tilted it so the brifters were at the lowest point and used about 1/4 can between the two brifters. Spray, actuate, spray, several time for each.