My comments are in-line below:

dpb wrote:
Does the chimney draw when the fireplace is in use?

It looks like it does normally. I lit some newspaper to get a smoky
fire going and the smoke went up the chimney. This was when it was not
windy though. I will try that test again when the wind kicks up.

If so, I'd not be
concerned. It's not possible to tell for sure, of course, whether the
situation is now different than before, but one possibility is if the
house is older and has been updated/remodeled that included
weatherizing so what used to draw well doesn't any longer. With air
tight houses, it's quite unremarkable that there isn't enough inleakage
to provide adequate draw. If you open a window a crack, does the
chimney then draw? If so, you've just demonstrated the above.

The house was built in the 60's and has been through a few major
earthquakes, so it's definitely not airtight. It passed inspection when
I bought the house in 2000.

A cap _may_ help a little, it may not. Their primary function is
keeping the weather and birds/vermin out, with a corollary function of
some spark collection and draft control. I'd probably add one if you
plan on using the fireplace simply to avoid the swallows building nests
(as they do in mine every year and every year I say I'll fix it before
next spring and forget until they've already got the nests established.
Maybe w/ your post reminding me I'll get the lift out and get up there
and do something about it this year!)

I see. I haven't had a problem with pests getting in yet, but it's
probably a matter of time before they do. If it doesn't hurt, I'll
probably put a cap in, just for that reason alone.