Does midodrine reduce amount of blood flow in the lungs at all?

No: It is a peripheral α-adrenergic agonist, finds use in the clinical management of patients with orthostatic hypotension or hypotension secondary to other clinical conditions or drug therapies. In patients with refractory orthostatic hypotension oral midodrine increases standing blood pressure and improves symptoms of orthostatism, such as weakness, syncope, blurred vision and fatigue.

Midodrine: Midodrine is an Alpha agonist but has relatively little data compared to most other approved drugs. It shouldn't reduce pulmonary blood flow but i can't find data to prove that.