I'll try to answer your questions:
1. end of June / early July can still be okay, although it's the beginning of Mauritian 'winter', it can be slightly fresh and you might have a bit more rain than in other months. However if you would like to travel around and get to know the island, and not just lie on the beach, it's an ideal travelling period.
2. Dangerous species are rare, but they exist: there are stone fish and sea urchins, the latter are not poisonous however if you manage to step into one and parts stick under your skin your holiday had been nice... stone fish are rare and hard to spot, however they exist, they are poisonous, and they are one of the reasons why you will never meet me in the sea without shoes... It might be different at public beaches, however I just don't take the risk.
3. Easiest or cheapest transport should be the question... easiest is by taxi. Cheapest is by bus. I would recommend a mixture of both. If you are travelling with a lot of luggage: take a cab. If you want to learn about the land and the people: hop on a bus.
I hope you will enjoy your holiday. If you should have any further questions I'm sure you will get answers here @info-mauritius.
Mme. Tortue (Mrs. Turtle)
DER WEG IST DAS ZIEL!