Learn Croatian: Ja vidim ovaj sir.

Vidjeti is a regular -jeti verb. To form the present tense, you drop the -jeti ending to get the stem (vid-) and then add the present tense endings. For 'I', the ending is -im, making it vidim.

Is the word ja (I) strictly necessary here?

No, you could just say Vidim ovaj sir. In Croatian, the verb ending (-im in vidim) already tells you who is doing the action. People usually only include ja for emphasis, like saying 'I see this cheese (even if you do not).'

The word sir (cheese) is the object being seen. Shouldn't it change its ending for the accusative case?

It is in the accusative case! For masculine nouns that represent non-living things (inanimate objects like sir), the accusative form is exactly the same as the dictionary form. If it were a living masculine thing (like a dog or a brother), it would get an -a ending.

Why use ovaj for 'this' instead of ovo?