1) Yes, as of now (v9.55.1) it's all at once. BUT, you can increase the issuance by issuing more later unless you "lock" the issuance.
2) So yes, there can be a custom distribution schedule. Example: start with 100, run a "stakeholder" vote via Counterparty broadcast or other method or decide by yourself as the issuer, then increase issuance to say 200. But, if you lock issuance, then it's impossible to issue additional quantity of tokens.
Another way of answering your rephrased questions:
1) Counterparty relies on the bitcoin blockchain (and the miners) and it doesn't have its own miners. The native currency (XCP) was created by proof-of-(bitcoin)-burn. Tokens/assets are created by burning a small amount of XCP
2) Coins (tokens/assets) can be distributed in any way you want, it's independent of pretty much anything - for example you could simply ask people to provide their verified address (which they could do in any number of ways that bitcoin supports, for example by signing a message) and send them some tokens. There's a faucet (google it) that works similar to bitcoin testnet faucets and you could use that as well.