Answer: All plants flower at one time or another. Virginia creeper or Parthenocissus quinqufolia is a large vine with un-noticeable flowers briefly early in the season. Later, it forms small fruits that the birds will eat. It turns a gorgeous red in the fall and is winter hardy to zone 3.
If you really would prefer no flowers, you could consider using an awning or perhaps shade cloth fabric over the arbor and then a few hanging baskets of foliage plants for decoration. Many houseplants will do well in a shady area, incuding philodendron and English ivy.
As far as spiders, I would not expect them to be any more numerous than normal -- it is outdoors, after all. Too, they can be usually be rinsed off with a spray of water from the garden hose.
I hope this gives you some ideas.
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