I know that “la boîte de conserve” is translated to “tin can”, however, that seems like a bit of a mouthful, so according to my dictionary it get's shortened to “la boîte de qqc” to refer to a “tin of something”, but I've also seen this referred to as “la conserve”, so is there a difference between “la boîte de conserve”, “la boîte” and “la conserve”? My initial thought is that “la boîte” is used when referring to a tin containing something and “la conserve” and “la boîte de conserve” refer to the actual tin itself, but I'm not 100% sure that this is correct, i.e. could you have “une conserve de qqc”?
Boîte is a generic term to designate a container in which things can be stored and if you use the word without specifying either the material or the content you do not know if you are dealing with the English "tin"/"can" or "box".
Une boite de gâteaux, une boîte à gâteaux, une boîte à cigares, designate the actual content (boite de gâteaux) or the intended content (boîte à gâteaux, boîte à cigares).
Une boîte en carton, une boîte en plastique, une boîte en métal, refer to the material of the container and in English you'd say "box" for the first two and "tin" or "can" for the boîte en métal.
Une boîte de conserve designates both the content (conserve meaning preserved food) and the type of container since it is only used for tinned preserves (and bocal for jarred preserves).
If you want to be specific about the type of preserve then, yes, you can replace the word conserve by the name of the actual food : une boîte de thon, une boîte de petit pois, and everybody knows you are talking about preserved food.
To talk specifically about the content itself without mentioning the container we can say : du thon en conserve, des petits pois en conserve, or even du thon en boîte, des petis pois en boîte.
Boite is more general it isn't always referring to a tin containing something. Boite equivalent to box. To answer to your second question, you could have a conserve of something.