品物 means stuff/goods/articles/whatever you use for some purposes. So an empty can isn't 品物 in a normal situation. It can be edible, though you usually use 食べ物 for edibles unless you see them as something you "use." So 品物 includes a present to your mother regardless of its edibility.

There are words for things for specific purposes, e.g., 贈り物 (present/gift). So things that can be 贈り物 are 品物 in some sense, but not all 品物 are 贈り物.

品物 is synonymous with 物品 (ぶっぴん), thought there is a slight difference in nuance. I think you can learn the difference through reading/listening the same way you learned the slight difference between "present" and "gift."

As for 商品 and 製品, their non-technical meanings in the vernacular are:

商品: things you sell. "Commodities" and its synonyms are very similar to this word in meaning.
製品: products you manufacture/manufactured to sell. Dell's laptops are their 製品, and your local computer shop sells them as 商品.

Your 製品 might be worthless crap if you look at as 商品 even when it's quite a gadget for geeks. But if your 製品 sells like crazy, it's a good 商品 regardless of whether it's a cheap knock-off made in China.

商品 includes produce as in "local produce" and "organic produce." One single apple can be 商品 too. Financial products such as derivatives are also 商品. Actually 金融商品 is the Japanese equivalent of the English word "financial product."

製品 often has a connotation that they're manufactured in a factory, plan, manufactory and so on. For this reason, some industries such as the cosmetics industry avoid the word and use 商品 instead when they refer to their products even when they're technically 製品.