Can seasonal allergies cause asthmatic symptoms? Will allergy shots help?

Yes to both. Pollens and outdoor molds can cause both nasal symptoms and asthma. Immunotherapy or allergy shots have been demonstrated to help both conditions. Controlling nasal symptoms can help with the asthma.
Yes and yes. The pollens that trigger seasonal allergic rhinitis symptoms can also trigger seasonal asthma symptoms. Thankfully, the allergy immunotherapy (allergy shots) that treat allergic rhinitis are also effective for allergic asthma. Allergy shots are a time-consuming commitment, but well worth the time and effort. An allergist can review benefits and possible risks.