This patient has major depressive disorder: loss of interest/anhedonia (need to have this or depressed mood),sleep problems, weight changes, decreased energy, thoughts of death. Meets criteria because > 2 weeks timeframe. SSRIs are first-line; paroxetine is in this category. SSRIs also help with weight gain - might be an added benefit if the patient is underweight.
The cardiac stuff might have just been a distractor, except that you probably wouldn't want to give tricyclics (i.e. amitriptyline) since they have pro-arrhythmic side effects. Patient probably has prolonged PR interval due to beta blockers.
Alprazolam is a benzodiazepine whose MOA is to facilitate GABAa action by increasing Cl- channel opening, this would be more for anxiety than the depression presented. Amitriptyline is a TCA that inhibits serotonin and NE reuptake and is not indicated for those over 65 (Beers). Buspirone is for generalized anxiety disorder and stimulates serotonin receptors, again this patient has depression. Carbamazepine is an anti epileptic drug that blocks Na channels and is first line for trigeminal neuralgia. Haloperidol is typical antipsychotic that blocks D2 receptors and should not be given to 65+ (De beers plus why would we want to give him extrapyramidal symptoms). Methylphenidate is a stimulant used in ADHD. Finally paroxetine is a SSRI that inhibits serotonin reuptake. It is first line for major depressive disorder and has fewer side effects than TCA.