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"There are six cats" or "I have six cats" would be
猫は六匹います・neko wa roppiki imasu・Cats (topic) six of them exist (animate)
六匹の猫がいます・roppiki no neko ga imasu ・ Six of them (noun-link) cats (subject) exist (animate)
counters are suffixes to numbers (like two slices, three sheets, four pairs, five loavesin English)
When "roku" gets combined with "hiki" there is some elision and with rendaku it becomes "roppiki"
When you place the number before the thing you are counting (second example) you connect that number with the object with the particle の, which is used to show possession or group of belonging. (It's like "six slices of bread)
"imasu" means "to exist" used with animate things (arimasu is used with inanimate). It also is used to mean "to have"
つ can be used as a general counter if you don't know the proper counter for a word. This counter uses the native Japanese counting system unlike most other counters. 六つ is pronounced "muttsu"
In certain circumstances you can use たち as a suffix for plurals (note this is typically only used with living things, even more usually only people, and roughly means "and company") 猫たち "nekotachi" would translate to "cats" but is more like "Cat and company" talking about a very specific group of cats, not just cats in general.
You'll see this with 彼女 - kanojo, "she" and 彼女たち - kanojotachi "they" (group of girls). And 私 watashi "I" becoming 私たち watashitachi "we"
Both が and は are grammatically fine, it only slightly changes the nuance.
は marks 'cats' as old information and would answer the question of "how many cats are there?" if 'cats' was already established/understood but the number is unknown.
が marks 'cats' as new information where you are simply stating there are six cats and could be an answer to "Do you have any pets?" where 'cats' specifically was unknown.
If you're trying to say "my cat", it would just be 私の猫 (わたしのねこ) without the は since it isn't a full sentence.
watashi no neko
watashi no neko wa kawaii desu.
My cat is cute.
"I" by itself is just 私, you only add a は when you're the topic of the sentence.
watashi wa neko desu.
I am a cat.