The problem is that none of the previous years' weather patterns for anywhere in the world can be depended on to be valid in subsequent years. GCT, and everyone else, can show tables of monthly highs and lows and rainfall, but you can't take that to mean it will be the same for your trip.
I took that Russia trip last August and for the first couple of days of the pre-trip extension in Kiev, it was very hot. And it had been very hot in Russia for a couple of months. There were also fires in Russia which caused a lot of smog. Two days before we flew to Moscow, everything changed. The weather was seasonal, the fires were out, the smog was gone, and they had the first rain in two months.
I would say to bring short-sleeved tops mostly with a couple of long sleeved ones, a couple of cardigan sweaters, a fleece jacket and a rain jacket with a hood. In other words, plan to dress in layers. Long pants, not shorts. Sensible shoes for walking on uneven pavements and cobblestones. Add a scarf or two and a pair of gloves. They don't take up any room.
It gets very windy on Kizhi Island, so be prepared for that.
You won't need anything dressy. Some people dress up for the ballet, but it's not necessary.