Any spellcaster can.
Dot.
Any caster can do research, no reason for druids to be an exceptioh
While a druid is able to do spell research keep in mind that as a prepared divine caster you already know all druid spells of the spell levels you can cast. That means the only reason to do spell research is to create a new spell. Creating a new spell is something that will require the GM approval. The spell is still going to be a druid spell so if the spell you want to create does not make sense for a druid spell research will not work. If all you are looking to do is to get a spell on your spell list from another class that will probably not work. If on the other hand you have a cool idea for a nature oriented spell that does not exist then you are probably ok. Again this will need to be approved by the GM.
Yes, but what you can do with that spell research is up to your DM.
A really nasty DM will let you spend fortunes and years in pursuing blind alleys in research, just as what frequently happens in real life.