Wilson's Disease: This diagnosis is highly likely due to the combination of symptoms presented, including tremors at rest, mild icterus (indicative of liver dysfunction), dysarthria, and the presence of Kayser-Fleischer rings (brownish rings in the superior corneal poles). Wilson's Disease is a genetic disorder that leads to copper accumulation in the body, affecting the liver, brain, and other organs.

Parkinson's Disease: Although less common in young adults, Parkinson's Disease could present with tremors at rest, bradykinesia, and rigidity. However, the presence of Kayser-Fleischer rings and liver dysfunction points more towards Wilson's Disease.
Essential Tremor: This condition typically presents with action tremors (tremors that occur when trying to use the affected limb) rather than rest tremors. The progression to involve multiple limbs and the presence of other neurological symptoms make this diagnosis less likely.
Anxiety or Psychogenic Tremor: Given the patient's history of feeling depressed and anxious, a psychogenic component could be considered. However, the objective findings of Kayser-Fleischer rings and significant neurological deficits suggest an organic cause.

Huntington's Disease: Although rare and typically presenting with chorea (involuntary, abrupt movements) rather than tremors, it's crucial to consider due to its severe prognosis and the presence of psychiatric symptoms.
Toxic or Medication-Induced Tremors: Despite the patient denying medication or recreational drug use, it's essential to rule out any potential toxins or substances that could cause similar symptoms.
Thyroid Dysfunction: Both hyperthyroidism and hypothyroidism can cause tremors, among other symptoms. Given the patient's anxiety and depression, thyroid function tests should be considered to rule out this potentially treatable cause.

Neurodegeneration with Brain Iron Accumulation (NBIA): A group of rare genetic disorders characterized by iron accumulation in the brain, leading to various neurological symptoms, including tremors.
Pantothenate Kinase-Associated Neurodegeneration (PKAN): A rare genetic disorder causing iron accumulation in the brain, leading to dystonia, parkinsonism, and other neurological symptoms.
Ceruloplasmin Deficiency: A rare condition that can lead to neurological symptoms similar to Wilson's Disease but without the liver dysfunction and Kayser-Fleischer rings, due to mutations affecting ceruloplasmin, a protein involved in copper transport.