Can the Planeswalker type be removed from permanents?
306.4. If a player controls two or more planeswalkers that share a planeswalker type, that player chooses one of them, and the rest are put into their owners' graveyards. This is called the "planeswalker uniqueness rule."


If you somehow remove or alter the planeswalker type of a planeswalker (Jace, Nissa etc are the types), you can play another Planeswalker with the same name yes.

In your example, the use of Song of the Dryads makes one of your Planeswalkers a Forest (it loses its types Planeswalker and [Name], and it gets the types Land and Forest). However, it also loses its abilities. I can't think of cards right know that just remove a type of a Planeswalker.

If it were to keep its loyalty abilities, it could use them just fine. The only restrictions for activating a loyalty abilities over the standard restrictions for activating abilities (costs, targets) is that only one of an object's loyalty abilities can be activated each turn.