Felt is a very durable fabric, and with gentle care, will last you for many years to come.
Wool is naturally water resistant to a great degree, so wearing a felt piece in the rain should not be problematic. It will have a greater degree of water resistance if it is a thicker piece that is more tightly-felted. A piece of laminate felt with a higher ratio of fabric to wool will not be water resistant.
Hand wash gently in cold or room-temperature water, with minimal agitation. Roll or press in a towel to remove excess moisture. For two-dimensional pieces, lay flat to dry. For hats, place on your head to regain the original shape, and gently tug or pull brim as necessary. Let the hat air dry in this shape.
Pilling is a natural characteristic of wool, and all wool will pill to a certain degree. This mostly happens in places with the most contact. Pills can be removed by gently shaving the piece using a razor or a shaver designed to remove pills.