Creating a transaction requires approximately 10 bytes to start with. Then, each unspent output that you use as an input to fund the transaction adds approximately 149 bytes. Each new unspent output that you create (each recipient that you send to) adds 34 bytes.
So if we send to multiple addresses one by one, each additional address will add ~10 bytes? For example, if we pack 100 receiving addresses into one transaction, then we can save ~1000 bytes of space?
Also, is there a limit for how large one transaction size can be (in bytes)? Or a limit of total receiving addresses we can include in one transaction?
Yes you would save about 1000 bytes of space.
And there is normally a mempool limit of 100KB and a maximum literal transaction size being 1MB.
You could contact a mining company who will probably be happy to mine your transaction if it's bigger than 100KB but it'll cost quite a bit in TX fees (viabtc allow you to broadcast transactions so they will pick up the transaction if they set rules that means it doesn't have to be on the mempool in order to be accepted). There was a block a while back which waas mined by F2pool I think that contained a transaction 900+KB in size.