What is the size of the set of spent key images in Monero?

What is the exact size of the set of already spent key images and how it can be checked? Do we need to download the Monero blockchain to get the size?

Yes, you'd need to download the blockchain. Once you have synced the blockchain, you can obtain the count of spent key images by inspecting the database like:

mdb_stat -s spent_keys ~/.bitmonero/lmdb/
...
Entries: 35830820

Using the is_key_image_spent api, we can check if a key image is a member of the set of already spent key images or not. Can this set be made a zero-knowledge set...

Key images already provide a form of zero-knowledge proof - proof that there is no double-spend without divulging which output is being spent. Each ring of outputs (so tx inputs) has a key image which is unique to one of the outputs in the ring but crucially doesn't divulge which is the actual output being spent.

1. Each transaction input has a key image, and there are around 2.1 inputs per transaction on average. So, the 'key image set' is around 2x the number of transactions on-chain.
2. I do not believe a zero-knowledge proof is currently available, although you may find the description of a hypothetical 'unspent proof' in section 8.1.5 of Zero to Monero 2 is in the right direction.