WyrWood

They are listed as Constructs yet under their description they receive none of the other benefits of being a full construct.
My question is do they get all the benefits as well as drawbacks of a full fledged construct?

Some of this is just assumed since it is part of the traits of the construct type.

For the bestiary entry, it lists 'construct traits' under immunities. For the race, it has construct as a type via the race builder, and that mechanic has a long section listing what essentially amounts to 'what is under the construct type'.

It doesn't have to explicitly spell out each and every single entry for the traits each and every time in each and every bestiary page or race page. Just listing the type does so- that is the primary purpose of types- a short hand listing of abilities shared by a group of creatures. It saves the game makers ink and paper not printing it out, and just making it a general rule 'check out the monster types'.

Construct races so far have been moderately powerful having some penalties added to them Android, and Warforge being the primary examples. Androids are powerful in a lot of ways but don't get any Morale bonuses which means most buffing spells don't work on them. While they are immune to exhaustion and fatigue they gain no benefit from raging since it grants a Morale bonus. With feats they can gain them but it's painful to attempt this.
Wrywoods are in a lot of ways OP. Think about this way. Point buy you have five not six stats to worry about. Assigned stats you still come out ahead since you can drop the lowest stat. Healing is the only serious issue they have and even then they still can at half rate. While this could be an issue that compared to all the other abilities a Construct gets seems minor. You play an Iron Priest from Technology Guide you heal at a full rate.
I have considered using the race for a character since I prefer races with a high Dex Int combination.

Construct races so far have been moderately powerful having some penalties added to them Android, and Warforge being the primary examples. Androids are powerful in a lot of ways but don't get any Morale bonuses which means most buffing spells don't work on them. While they are immune to exhaustion and fatigue they gain no benefit from raging since it grants a Morale bonus. With feats they can gain them but it's painful to attempt this.

Wrywoods are in a lot of ways OP. Think about this way. Point buy you have five not six stats to worry about. Assigned stats you still come out ahead since you can drop the lowest stat. Healing is the only serious issue they have and even then they still can at half rate. While this could be an issue that compared to all the other abilities a Construct gets seems minor. You play an Iron Priest from Technology Guide you heal at a full rate.
I have considered using the race for a character since I prefer races with a high Dex Int combination.

How does a wyrwood heal at half rate? Or any rate for that matter?

They don't get HP back from resting and Cure X Wounds spells don't work on them (cure wounds spells only doing half effectiveness is an Eberron warforged thing). Iron Priests can channel energy to heal wyrwoods and other constructs/robots for half effectiveness. Make Whole is too high a level for level 1 wyrwood creatures to benefit from and only level 1 fast healing spell available is infernal healing (or the mechanically inferior celestial healing which is actually available to good clerics) and its 1 round casting time ensures one doesn't wanna use it in middle of combat unless it's really needed.

How does a wyrwood heal at half rate? Or any rate for that matter?

They don't get HP back from resting...

I believe there is no rule that directly cites "nonliving creatures don't heal by resting"

Edit: Apologies, I didn't see this line:

Cannot heal damage on its own, but often can be repaired via exposure to a certain kind of effect (see the creature's description for details) or through the use of the Craft Construct feat. Constructs can also be healed through spells such as make whole. A construct with the fast healing special quality still benefits from that quality.

Half rate was assumed using War Forges as an example. Still not a bad race considering. Although prefer conversions of War forge and androids over this race the rules regarding them more detailed making confusion and questions less. Using point system rule these guys are seriously higher then most other races including a noble drow.