concerning typhoons - yes. if you mean monsoons, also more or less yes, although the amihan in particular (northeast monsoon) seems significantly stronger, considering the open ocean just to the east. southern leyte is generally a little cooler (such as it is), and a little wetter. dry season also seems to arrive a little later, but as a rule if cebu is affected so is southern leyte, both being the same latitude and only a stone's throw away.
as to nightlife - very nearly "wala". there is the moose and squirrel, aka jerry and alma's, a 5 minute walk up the road. food is good, you can get a cold beer, and you will be treated well. budget in the order of 250 php per meal, a little more if there is fresh shrimp or a nice piece of fish available for the grill, plus a beer 60/70 php. you can up that with a couple appetizers or kinilaw. padre burgos (poblacion) has a couple places, frequented by locals and expats, where you can have a drink - a beer, a rum, and if you are lucky maybe there is a little vodka or gin hanging around. backpacker budget. bring your own quality brown liquor if that's your interest. the closest place to get it is cebu city, or tacloban on a good day. as the trikes and jeeps don't run much after sunset, you will walk the 2 km back to peter's unless you have wheels. southern leyte divers has good food and maybe the best bar in all of leyte in a nice setting. budget for flashpacking, but without your own transport you will be walking the 4 km back. haven't had a meal at sogod bay s.r. in years, so can not comment.
if you are really desperate to see what night life is like in a provincial capital you can go on to maasin, but i'm not sure why you would want to. at about 25/30 km away you will have a long walk back, but there at least it is possible (usually) to find someone willing to throw you in a trike or chartered jeep or whatever happens to be rolling down the road and get you back. unless you absolutely got to do it, driving more than a few km at night is not highly recommended. it's a jungle out there. most people are happy with the food and camaraderie at peter's, and for good reason, and with wandering up the street a bit for a change of pace.