let's start with das. Normally, it's just an article for neuter words, like "das Auto". But if you use it without any noun it refers to, it becomes a demonstrative pronoun:
Das Auto ist schön - the car is nice. Das ist schön - this is nice.
Since it doesn't refer to anything in particular, it doesn't get any flection. Whatever you currently do or see or experience, it is nice.
Now if you want to talk about a specific noun and use a demonstrative like "this car", you use dieses. Since it refers to a noun which has a gender and also a case, you have to decline it according to the noun:
This car is nice - dieses Auto ist schön. (nominative sg neuter)
This dog is dangerous - dieser Hund ist gefährlich. (nominative sg masc.)
This cat is black - diese Katze ist schwarz. (nominative sg. fem)
These children are loud - diese Kinder sind laut. (nominative pl)
I like this dog - ich mag diesen Hund (accussative sg. masc.)
and so on. You can look the tables up. You can actually drop the noun here, but you still match the pronoun to the item you refer to:
Dieser Ring gefällt mir - I like this ring.
Dieser gefällt mir - I like this one.
You mean a specific noun without naming the noun, so this one is a pretty good translation in my opinion. Compare to unspecific "I like this" - "ich mag das".
So far is what you would probably use in writing. In speech, you sometimes use the article as a demonstrative, but then it gets a flection, too. You have to lay heavy emphasis on the article to convey that you use it as demonstrative pronoun:
Der Hund ist gefährlich - the dog is dangerous.
Der Hund ist gefährlich - this dog is dangerous.
again, you can drop the noun but keep the declinated form and the emphasis:
Der ist gefährlich - this one is dangerous.
At last, we have dies without any ending. You can use it the same way you used das as a demonstrative in the example above:
Das ist schön - this is nice. Dies ist schön - this is nice.
It sounds a little more stilted. You would mostly hear it while someone points a finger and explains something, but using das is correct then as well, so you can easily live without dies.
There are some more obscure demonstrative pronouns in german, like the aforementioned jener, derjenige, derselbe and solcher, but you don't need to worry about them for now.
Demonstrative pronouns can be nominative, accusative (what), dative (to what) and genitive (of what). This means there are 16 versions fortunately (if you consider the mas,fem,neutral,plural) of dies. Here the point is to practice the nominative ones that can be M: dieser, F: diese, N: dieses, pl: diese depending the gender of the thing you are talking about.
Dieses is 'this' - Das is that
you can look here for how to use Dieses: http://deutsch.lingolia.com/en/grammar/pronouns/demonstrative-pronouns#comments
Yes, "This child is eating fish" is also correct. Unlike English, Standard German does not distinguish between the simple and the progressive aspects. Standard German doesn't have this grammatical concept at all. Thus, a sentence like "Dieses Kind isst Fisch" can mean both "This child eats fish" and "This child is eating fish".