I can't speak to the weather at that time from personal experience, but the region is famous for being rainy, cloudy, and did I mention rainy, for most of the year. I was there in late May and had great weather, but the locals made clear that this was not the norm, even that close to the official start of "the season" on June 1. The season ends mid-September. Note that many things are closed or have shortened hours in the off-season.
Of the sights you listed, the only one I went to is the Vizcaya Bridge. I loved it. It was the first transporter bridge built, and one of the few still in operation. If you have any interest in engineering, it's great; otherwise, you may not be as excited as I was.
Here's what I wrote about it at the time:
"Next on the agenda was the Puente Colgante at Portugalete, a half hour north of the center. This means "transporter bridge," and it was the first one in the world and one of the few still operating. If you don't know what a transporter bridge is, it's easier to show than to describe. Fortunately, the opening sequence of the movie The Young Girls of Rochefort takes place on one. So, if you haven't seen the movie, watch this clip. You may, if you wish, ignore the dancing - just pay attention to the bridge.
You see, it's a bit of the roadway that is suspended from a great height above the river, and moves across it. This means that boats can go under the high part unimpeded, but there need not be a very long approach as would be needed with a high span of a regular bridge. The director of Young Girls of Rochefort, Jacques Demy, was a bit obsessed with these bridges (there had been one in Nantes where he grew up, but it was dismantled in the late 1950's), and put them in several of his films one way or another (in a movie set in Nantes, he recreated it in the background of the credits with special effects). Rochefort's transporter bridge still runs, but I'm not likely to get there, so this one near Bilbao was a must-do, especially when I learned it was the first one ever. For €0.40, you can take the transporter across; for €10, you take an elevator up to the top level, walk across while listening to an audioguide all about the bridge, and then take an elevator back down to take the transporter back across. Needless to say, I did the whole shebang. I don't recommend it for anyone with any issues with heights; for those people stick to the transporter part, which is only a bit above the river (remember, the same height as the surface roads on either side).
By the way, I was amused to see the signs in three languages (Basque Castilian English) on the transporter saying Do Not Smoke and Stay In The Car. So, you can't recreate the movie's opening scene yourself - at least in Bilbao - without getting in trouble."