Answer for question 2: Do the medicines have any connection to Dr. Manhattan's cancer?

I start with this question, because here we know the answer from the comic and the movie.

It is certain that the medicines have no connection with the cancer that Dr. Manhattan supposedly caused.
Because as stated in the comic and the movie, the cancer never originated from Dr. Manhattan.
Everyone from Janey Slater to Moloch was exposed to radiation by Adrian Veidt;
in a plot to cause the "too powerful and unpredictable" Dr. Manhattan to exile himself to Mars.
However the Comedian wasn't exposed to this radiation by Veidt, so that is ruled out.

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Answer for question 1: What are these medicines?

The medicines are actually never specified, not that important to the story and most likely just there to show the broken state of an aging man.

So any answer would be based on speculation, but we can give a few possible reasons for those medicines:
(For a quick reference I put the types of medicines in bold)

• Medicine for old age:
  The Comedian was 61 years old at the time of his death. An slightly advanced age, so this might have been some kind of blood pressure or cholesterol medicine. However any information found in the comic or movie states that he was in terrific shape for his age. This is also apparent in the final fight by rolling, jumping and ducking quite fast and being able to punch through a brick wall.
  Probability: 10%

• Painkillers:
  In his lifetime, working as a vigilante in The Minutemen (1939 - 1940) and the Crimebusters (1966 - 1977) and later also as a soldier for the United States (1945 - 1985), he has probably suffered quite a lot of physical injuries. So the need for painkillers to manage and treat these sustained injuries is not so far fetched.
  Probability: 25%

• Psychological drugs:
  Edward Blake was a troubled youngster with rages of violent behavior and as a young vigilante, he fought most of the gangs of New York in a violent fashion. Then later as a soldier/government agent, he served in World War II, the Vietnam War and countless covert ops. So he has certainly seen and done a few things which could cause PTSD in an ordinary man. Edward Blake however is no ordinary man.
  He did however make a discorvery on Veidt's island that caused his practised cynicism to be shattered. Resulting in a broken world view, an identity crisis and sleepless nights. This is made apparent when he visited Moloch in a panicked state.
  It would be very possible that he took sleep, anti anxiety or anti depressant medicine to cope at this point.
  Probability: 50%

As you can see it remains a guess. But with the evidence at hand, it is at least a calculated guess.