Hashing large messages, how does it work? Is the message divided into chunks, each chunk hashed separately, or something else?
While hashing, the messages are divided into chunks and padded so that a message is fit into the chunks, properly. For example in SHA-256 the message appended 1, then many zeroes and finally the message size in 64-bit added. The total size must be $x$ multiple of 512 where $x$ is minimum.

No, they are not separately hashed, they are chained. If not it how would you combine the output? If you don't combine what is a hash there? There is actually, hash trees famously know as Merkle Trees used to verify data blocks, or many files where you only keep the root hash.

SHA-256, for example, uses  Merkle–Damgård (MD) construction in which a compression function $F$ which is

$$F:\{0,1\}^{256}\times\{0,1\}^{512}\to\{0,1\}^{256}$$

The 512-bit input space is the message space. 256-bit input space is the previous hash, if it is the first block then an IV is used.

let $$M\mathbin\|pad=m_1\mathbin\|m_2\mathbin\|\ldots\mathbin\|m_\ell $$ $$H_i = F(m_i,H_{i-1})$$ with $H_0 = IV$ then

$$\operatorname{SHA-256}(M||pad) = F(m_\ell, F\ldots F(IV,m_1)\ldots))$$

This tells you about the MD construction, MD5, SHA-1, SHA-2 uses MD. There is also sponge construction that SHA-3 is based on, which left the OP as an exercise.