Cycling-tour through Croatia – bike allowed on board ferries in May?

I am planning a cycling-tour through Croatia in May and want to ask you – are there any ferries going in May?

Yes, ferries are running in May.

Ferry from Rab – Novalja May 22nd and is it possible to bring a bike on board?

Rijeka to Novalja is foot passenger only ferry run by Jadrolinija and to the best of my knowledge they dont take bike on board (they might take bike in bag, you would need to check with them directly)
Use instead Stinica to Misnjak car ferry : https://www.croatiaferries.com/stinica-misnjak-ferry.htm from where you can reach Pag island on your bike.
See local north adriatic ferry map: https://www.croatiaferries.com/maps/north-adriatic-local-ferry-map/

The ferry from Dubrovnik to Bol – is it going in May (or June), too and is a bike allowed on board?

Take instead Dubrovnik to Milna (also Brac island) ferry, going from May to September, they take bikes on board: https://www.croatiaferries.com/dubrovnik-milna-ferry.htm ; book it at the orange button