I will start by working backwards. The warmest month, statistically is July followed by August. This can be seen on our Normals, Means and Extremes page (http://www.mountwashington.org/weather/normals.php). It should be noted though that this page is currently under an update, so some of the numbers are a bit dated.
You can also check out our WS Form F-6 page: http://www.mountwashington.org/weather/f6/ which holds at least 7+ years of data that we have made public out of our 80+ year history. On this page, just click on any given month and you will see the past highs, lows, averages, etc for each month and day. This is very useful in planning as the best way to know what the weather will be in the future is to look at the past. But, with that being said, and like whitemtnvisitor mentioned, just because we are statistically warm and dry on any particular day, it will not be like that every year. And just because it is hot and sunny in the valley doesn't mean you will encounter similar weather as you ascend. Always pack for the unexpected even if the forecast is calling for great weather. I am talking from experience, that just because I and/or other meteorologists may call for awesome weather, on occasion, as you may know, we are not exact. So, always pack some extra gear to stay warm or dry in those one off occasions. It is easy to pack a layer or two for those just in case moments in life. Think of it like carrying jumper cables in your car. Most of the time you won't need them but it's safer to have them than not; same as warm/dry clothing.
As far as precipitation goes, I can't readily look that up as I would have to think about how to do that inquiry in our database. But I know in my time here, the longest time without precipitation of any form that I have experienced is about 8 days. I am sure, looking at our monthly means and extremes though, with some of the minimum months we have my guess would be one of those months holds the record. But, New England is called the tailpipe of weather for the US, so that means if something forms anywhere in the lower 48, odds are at some point it will impact us. Unfortunately though, there is no rule of thumb like if it is raining today it will be dry and clear for three days after. Nor is it similar to conditions in the west that fit a Gaussian curve where summer is nearly bone dry and all the precipitation comes down in the winter; precipitation in New England is a year round affair as you can see from the normal totals on the page referenced above. While some months during any particular year, it rains every single day, other times we'll just get bursts of rain then nothing for days. So, this stat will not help in planning any hikes per say. I think the better thing to look at is our snowfall stats since snow takes a whole different set of equipment and mind set than just rain would. The least snowiest months are July and August with a mean of a trace for July and 0.3 inches for August. That's not to say it won't snow more than those numbers, but on average, these would be the least likely months to have to worry about snow and ice on the summit. But like everything else I keep driving home, it still can happen and you should be able to dress for those events. But, statistically, when telling my friends when to hike this mountain, I will tell them to aim for July or early August to get the best and safest possible conditions. And if your plans don't pan out, go as far as is safe and try again another day. The mountain's not going anywhere.
Staff Meteorologist/Night Observer, KMWN (Mt Washington Obs., NH)