Hi Benjamin, thanks for reporting. For the next time, you can take a look at this [template][1].
And in [this other forum][2] we have many reports you may want to take a look at or comment.
In general, we prefer separate bug reports, which allow people to comment on a specific bug. But many people post lists of bugs, at risk of being closed or changed 
Concerning the bugs you mentioned:
-What release did you play? The latest one?
-Building the houses: I’m pretty sure that now they are capable of finish a house, unless you’re referring to the roof, which is not still fixed (but the Team is going to work on building pretty soon).
-Fence gate: I think someone mentioned that the animation wasn’t working well, or that the characters could pass through it.
-Chopping trees: it depends. If the tree is in another terrain level, they can’t chop it (for the moment). If it is too far on the same level, it only takes time for them to reach the tree, and may get distracted by other events on their way there (hunger, sleep,etc).
-Carpenter: the carpenter only does carpenter tasks, and if he gets distracted by fire, hunger, or other things he may forget his tasks. But with the latest releases every settler is able to resume their work, which is what really matters.
-Trapper: yeah, the trapper is causing some important bugs. Probably someone has already reported them here or in the other forum.
Well, that’s pretty much it. Thanks again for reporting. 
[1]: Welcome to the Bug Report Category!
[2]: http://www.stonehearthguru.com/forums/alpha-beta-testing-nexus/support/bug-reports