Start Dubrovnik or Zagreb next May for month

Not too cool or too early for Slovenia. I've been there twice in May, including this last May. I've had some rain each time, but I've also had rain in September there. I think they get their share of rain. It certainly wasn't cold in May - I had no trouble driving up through the mountains by Lake Bled. There may have been snow up in the Julian Alps (didn't go up there this time), but I saw no snow at all at the higher elevations where I drove.

You never know with the weather in May, but I'd probably be inclined to start north and finish in the south in Dubrovnik. You're always playing the odds with the weather. I was lucky two years ago driving from Rovinj down the coast to Dubrovnik and Kotor - I had only a tiny bit of rain here and there. Mostly good temperatures but also saw only one sunset into the ocean (in Rovinj), which was a disappointment, because I've seen the sun set into the Adriatic before in Croatia (in October), and it can be magical. Next May could be rainy all month - who knows?

I haven't done any cycling in Croatia - can't help you there.

I spent a few hours in Sibenk and loved it - considered spending the night but headed south to Trogir instead (I had an itinerary with a lot of places on my agenda). Consider a side trip to Krka National Park nearby from there - a huge cascading watefall is the highlight, bigger than any waterfall at Plitvice (which I still prefer to Krka though).

Some ferries and catamarans run year around along the Dalmatian coast. There are probably more crossings in the summer, more options, but you can certainly plan a doable itinerary in May with some island connections. For example, I know there are year-round catamarans (weather permitting between Split, Hvar, and Korcula, and there's daily bus service between Korcula and Dubrovnik (uses a short ferry over to Orebić). I think as long as you aren't set on a fixed island itinerary and can be flexible, you can certainly figure something out.

You can find 2017 ferry info by googling - that's how I've always done it. You won't know exact schedules for 2018 until 2018, but you can guess they'll be similar to 2017. Again, just be flexible there, pencil in a few of the easy islands like Hvar and Korcula and figure out the rest closer to arrive. May isn't high season so it shouldn't be too hard to juggle things around.

I've been to Croatia a few times in September when it tends to stay warmer longer in Dubrovnik than in the north. I presume the reverse would be true in May, so I would opt to travel south to north if planning a May visit.

The area south of Split has good bus and ferry coverage, so I would save the car rental for when you depart Trogir/Split and travel north. Having a car makes it much easier to visit Krka, Plitivice and Istria or to make a swing through Slovenia on your way to Zagreb. I rented a car in Ljubljana and was allowed to drive it into Croatia and back, so I presume the reverse is also OK, especially now that Croatia is part of the EU. There are some very nice all-day tours from Dubrovnik to Montenegro, so I would look into those for your Kotor visit. For Mostar, I took a bus from Dubrovnik, stayed overnight and then took a bus next afternoon to Split, which worked well.

Catamarans to the islands will operate on low-season schedules in May, which means they may not run every day of the week. The slower car ferries tend to run all year long, but there may be fewer daily sailings in low-season. Jadrolinija (ferries and catamarans) and Krilo (catamarans only) are the two boat services along the coast, with Krilo services skewed to the southern routes. Split and Dubrovnik tend to be the best starting points for island visits, although there's also bus service from Dubrovnik to Korcula. Note that it's tricky to connect Korcula and Mostar using public transportation.

Edit: When I rented a car another time for use only in Croatia, Uniline offered one with no drop-off fee provided I rented for more than 3 days. I was happy with the car and their service, which included free pick-up and drop-off near my hotels. I believe Uniline is now part of Europcar.