Because adjectives have to agree with noun, "kot" is in accusative so the adjective have to be in accusative too.

You could analyse it like that:

• First, you establish the subject(which is 'I' here) – since it is the subject, you need nominative and English 'I' tells you that you need 1st person singular; so "Ja"

• Since you know the subject, you can now deal with the verb – conjugation of "mieć" tells you that you need "mam"(1st person singular present tense), so you end up with "Ja mam"

• Now you need the direct object, the thing that subject possesses – excluding some exceptions(verbs that need different case; prepositions and so on) direct object is marked by accusative in positive statements(and genitive in negations), checking "kot" you get "kota", so you end up with "Ja mam kota".

• Finally, you need the adjective to describe the cat – you can go two ways here; You can simply check the declension of "czarny", knowing that "kot" is masculine animate you go with "czarnego" and knowing that adjective in front of the noun describes particular trait of the noun while after noun, adjective describes wide category/class to which noun belongs, you end up with "Ja mam czarnego kota".

• Alternatively, if you know "czarny" and don't want to check it's declension, you can get to it analytically – question for adjectives is "jaki" and for masculine animate in accusative it is "jakiego", so you take it's ending(jakiego) and apply it to the stem of czarny (czarnego) and you end up with the same "Ja mam czarnego kota". Obviously, for this method you need to get the hang of -y/-i alteration(jaki? czarny; jakim? czarnym but jaki? krótki; jakim? krótkim) but after a while you should get a hang of it.

Note that Polish actually has such question words working as guidelines to declension for everything and in order to internalise declension patterns you will have to learn them, but 1)it is not needed if you do not plan to learn Polish really well(B2/C1 or higher) and 2)most of them are pretty complicated, due to irregularities which is not the case for adjectives(there is only one regular pattern for those and the only problem is the -y/-i alteration), so they are great place to start learning them – which is also the reason why I mentioned it here(and left out the nouns and verb too ;-) ).

Adjective also declinate. „Czarnego” is „czarny” in accusative case.

I don't think we have -ch suffix, there is always some vovel before,

I assume you are asking about difference between czarn-ego and czarn-ych

-ego is genitive adjectives describing masculine and neuter nouns, and accusative adjectives for masculine nouns that have genitive=accusative, or their own accsative form

-ych/ich is plural - all genitive, all locative and masculine personal accusative.

so

Mam czarnego kota (accusative) (I have a black cat)
Nie mam czarnego kota (genitive) (I don't have a black cat)

Mam czarny ołówek (accusative)(I have a black pencil)
Nie mam czarnego ołówka (genitive) (I don't have a black pencil)

Mam czarne koty (accusative) (I have black cats)
Nie mam czarnych kotów (genitive) (I don't have black cats)
Mówię o czarnych kotach (locative) (I talk about black cats)

Mam czarnych kolegów (I have black friends)
Nie mam czarnych kolegów (genitive) (I don't have black friends)
Mówię o czarnych kolegach (locative) (I talk about black friends)

What is meant is just "I have a black cat". Which will be the default English sentence now.