1: The weather will be similar, probably slightly warmer in June (which is a good thing), but also more visitors (if that bothers you).
2: Three weeks is plenty of time to explore Lofoten on bike. A car would not be required.
3: If flying in/out via Evenes (Harstad/Narvik) airport, it would be a good idea to use the bus until you reach Lofoten, to avoid biking along the the main road (not too interesting, and some long tunnels on the LOFAST road.) If you go by plane or train to Bodø, use one of the ferries from there. If flying to one of the tiny, local airports in Lofoten (Svolvær, Leknes, Værøy or Røst), just bike from the airport.
4: I biked there - I live nearby. :)
5: As mentioned, three weeks is plenty for Lofoten. I would advise venturing into the Vesterålen islands north of Lofoten as well, they are just as interesting, less road traffic and more secondary roads suitable for biking. In Lofoten, you'll have to bike many of the stretches both ways, there's few "loops".
6: You'll probably be fine making reservations as you go - call to the next place in the morning. Fully booked accommodation could be an issue in June, less so in May. I would highly recommend that you bring camping gear, as this will enable you to stay in many nice places where there's no lodging, and also take away the stress of planning. Getting a shower a few days a week is nice, though. :)
