1) I'm concerned about the condition of the secondary roads in June. Are they passable in a 4x4 during the rainy season? We'll be there between June 30th and July 15th. Will be renting a 4x4 Toyota Hilux Pickup to get around.
2) Particularly interested in nature and birds and places off the beaten tracks and outdoor activities. Also interested in staying at small, clean lodgings near the Arenal Volcano. Will also be checking out the Lonely Planet. Any additional tips for lodgings?
Thanks

It just depends.
The two main reasons roads become impassable are washouts and high rivers. The high river levels are predictable and you can count on not being able to cross many fords along the monkey trail up the south and west coast of the Nicoya peninsula because of them.
Mudslides are less predictable and can take out paved highways as well as secondary roads.

A HiLux is a true truck, and should be able get almost anywhere, as you will need the LOW gear for the sloppy rutted roads, more in the mountains, and the coastal roads that few people use.
A word of warning, rivers can be a real dangerous obstacle, so walk the river first, and the opposite river bank, to be sure you know what depth the river is, and also how much speed you need exiting...up and over a embankment.
Most rental agencies do NOT allow you to drive on beaches, and also cross rivers...so read the fine print, as $1000 deposit is required.
Dont go where you dont know....you can end up on a desolate road/trail, and slip down a embankment and injured, some ravines on the roads are a few hundred feet down, so keep in mind, no one will know your down there...and climbing up unjured would be a ordeal, injured, almost impossible.
Snakes should be a huge risk when off the beaten path, do not just meander in the bush/jungle or canopy....deadly Fer da Lances are everywhere, have a sort fuse, and will strike without being provoked.....

Accomidations around Nuevo arenal vary from $30 ish dollars to $150 dollars usa. If you are looking affordable I stayed at Alpino Cabina next to the closed hotel River Rock. There is a place a 15 min walk great food/coffee and view of the lake called Cafe y Macademia. It's new and not in the book. However, another place I stayed in Nuevo Arenal is La Selba it is named after a 500 year old tree which was named tree of the year. Amazing and you need to see it even if you do not stay there. That place is very nice Alpino is a cottage with a kitchen so I have small kids and that worked out better for me. I was driving in to Nuevo Arenal from Tiliran direction.
Have fun!