Please help me identify these two rocks.

[Velodromed]

We find a lot of fossilized wood here in Texas, but as I learn more I’m discovering some may be coral instead. The rock in the first four pictures below I assumed was petrified wood, possibly palm. But as I learn more, it looks like it may have some coral like features? 

 

The second couple of pictures are of a light weight rock we’ve come across a few times. Out of curiosity, I cut one in half and cleaned it up with high strength vinegar followed by CLR. I now can see what looks like a branched chalcedony vein running through it and a druzy or two, I believe they’re called. Is it volcanic? A fossil? Or just a kinda cool rock…

 

My wife, 10 year old daughter and I got into rock polishing a year ago. Now we’re trying to learn about fossils because we’re constantly running across them. I appreciate any opinions on both rocks as it’s helping us learn. Have a great week everyone!

[Rockwood]

I see wood here, but I'm confused as to how the other textures relate to it. 

[Brandy Cole]

Your first four pictures do show petrified wood. I'm not familiar enough with petrified palm to confidently say, but I think it could well be based on the speckled end-piece features.

 

Also, rocks aren't my area, but it looks like you may have a calcite vug.  I wouldn't call it a geode because it's not the rounded fully hollow shape I associate with those.  But it's still a really pretty and interesting find.