Is Singapore a good place for a student looking to study and explore Asia?

I don't think you will have problem filling up those 5 months weekends.
It's all depend on your interests, of course, but there are lots of places nearby, tourist attractions (gasps!!!), etc, that can easily take whole day itinerary.
Kusu, St Johns, Pulau Ubin are some of the islands nearby that you can explore.
Theme Parks, Zoo, Botanical garden and other natural parks you can lazily spend your days on.
Plus your side trips to other neighboring regions, you wont have enough time as it is.
Don't forget to do your school assignments from time to time.

Yes it is. As long as you can afford it.

I cannot speak for the entire singapore population on what we do during the weekends, but there is a lot of things to do in Singapore.
Soak up the limited nature Singapore offers, if that is not your thing, lines of pubs and clubs in Clark quay await you. Or you can go on a food conquering adventure!
Despite living in Singapore all my life, I do believe I haven't try all the (famous store)food yet.. I know I'm ashamed of myself too. But I digressed.
Besides, who can say no to food.
You can visit museums, USS and plenty more.
Give Singapore a chance, we are not as boring as people say we are.

Besides the islands mentioned above, there's also bintan and batam(islands of Indonesia). They are approximately an hour away from Singapore by ferry. We do go there during the weekend sometimes to get away from the city life(by lying on the beach and do nothing all day long, but you can do that in sentosa too).

I hope I help. :D

#3:
To that I would have to ask: are most in Singapore merely passing through(meaning that it is essentially a weigh station for those looking to make money and move on to where they are from) or does most of the population live most of there lives in Singapore?

think approx 60% belong to the latter right now. it's changed a lot, increasingly difficult to recognise/feel at home in the country that one grew up in. in the past 10 years i've had to speak a lot more Mandarin instead of English in daily life - the proportion of people here who don't (in some cases, refuse to) speak English has increased. whenever i land at Changi i don't feel like i'm returning home, but 'returning to office'. which department/faculty will you join at NUS?

#9:
So, I have been wondering what Singaporeans typically do with their weekends?

eat shop movie. overtime work/weekend duties. pack their kids off to extra classes.

What is a typically night on the town like for a Singaporean?

eat shop movie. some drink. for an increasing number - pop out of the office to get dinner before returning for overtime work.

and lastly, are the islands surrounding Singapore interesting to explore?

Ubin is good, Sentosa if you like 'manufactured entertainment'. many of the islands are off limits - military training/live firing areas/ammunition dumps, landfill for incinerated garbage, oil refineries & other petrochemical industry uses. but it's easy to get a boat from Changi to places like Penggarang in Johor, Malaysia.