about counters

Hello!

I learned that when I want "to count" something I can use two kind of structures.

I would like to have two apples.

1st ex: niko no ringo wo tabetai desu.
2nd ex: ringo wo niko tabetai desu.

If this sentences are correct, is there a difference in their use? Or one is better than the other?
I know that there are very simply example, but at my level I can't do more!
thanx.

The second sentence is the correct one.

The first one "might" be used in particular circumstances where the number is crucial, but I cannot come up with an example of the situation.

Ok.
If I can't remember the specific counter for a kind of thing, can I use the generic in every situation?

Ex:
私のうちにねこがみっついます。
instead of
私のうちにねこがさんびきいます。

Can I also say "watashi WA uchi ni neko ga sanbiki imasu" and get the same meaning?

If I can't remember the specific counter for a kind of thing, can I use the generic in every situation?

Not really. That generic one is for inanimate things, so you should remember the one for people ("-nin"), animals/pets ("-hiki") and the generic one at least. With that you can get by.

Can I also say "watashi WA uchi ni neko ga sanbiki imasu" and get the same meaning?

Almost the same meaning, but not quite. A situation I could imagine anyone might use that is, let's say, a group of people are talking about what kind of pets everyone has, one says "I have a dog," another one says "oh I have two guinea pigs," etc., and now it's your turn. So you would start out saying "When it comes to ME (as against others), then I have three cats."