The 2nd lake was looking quite dry in late June
That wasn't my impression. I was there the same day as you, and it looked the same as it has the other two times I've been up to Shepherd Pass. I think it is a year-round water source, and I agree that it is better than the second lake.
Other places to camp on the approach hike in are at Mahogany Flat and Anvil Camp. Anvil seems to be especially nice because is about 2 miles from the top of the pass (2 miles in) and there are quite a few sites and a nice stream to get water from. Also, the first lake at the pass is generally prefered because it offers some protection from the wind and it's really unnecessary to gain the extra 200+' up to the next lake. The 2nd lake was looking quite dry in late June so I wouldn't count on it being a reliable source for water later in the year.
Just a wee question on the name of that mountain. If it was named after the same John Tyndall, the scientist who's name was borrowed for our national scientifc research institute where I work (www.tyndall.ie), then the guy was Irish, not English!! More details on him can be found here: http://www.tyndall.ie/contact/tyndall.html
Only bothered seeing as the name is very familiar to me :)
A flash flood last summer (2013) washed out a section of the Shepherd Pass trail at approximately 10,000', just below Anvil Camp. It is necessary to hike approximately 100 feet upslope from the trail to cross near a large rockfall. There is a second washout a little higher as well, but is easily bypassed and is cairned.
There are no longer restrictions on when you can visit the area. The trail has been rebuilt to get around the big washout below Anvil Camp.