Given that there are several somewhat unusual symptoms going on it would be worth a chat with the doctor. A hormonal imbalance issue can cause many of these things. Cold hands and feet can be thyroid or poor circulation due to a heart defect. Do watch him for developmental delays or that he is not progressing on the normal path - ie getting too far ahead in one area but lagging way behind in another. Given the premature start it will take him longer to catch up.

For the nose issues, an allergy comes to mind. Is he exposed to cigaratte smoke in the home?

On the food again the imbalances can cause the brain not to register when they are full or an agressive metabolism could mean things are going through the system too quickly. Word of caution if that is the case and that is that the child won't be getting all of the nutrients from the foods because there isnt' time to convert them so may be lacking in things like iron, etc. - all of which helps with brain function so it really is a vicious circle that only a doctor can diagnose and treat. Hopefully if you mention some of this to mom and put it with the idea of everything together and how the child could be missing out on needed nutrtion and therefore learning development she might be more likely to take him to the doctor than over the nose. Let the doctor discover the nose when he is there or have mom mention it as part of the feeding issues - allergy. If she then gets to see a nutritionist they will go over everything she is feeding/not feeding at home too.

Is mom feeding enough breakfast and supper or is he trying to make up for it during his time with you.