Should I pull out the tree? It stands at roughly 10 feet tall. With green colored branches. The stem of the tree is indeed brown and wooden. I have no clue how deep the roots are. Will the roots damage my home?
Well...don't get rid of it. Won't hurt anything this year. I would dig a trench about a foot deep now, around it's 'drip line' and fill it with straw. This winter I would dig it out and move it to a better location. You could even wait until next year, keeping the roots trimmed and growing within it's new 'root ball'...make sure that you keep this root ball well-watered.
Don't do this when the ground is frozen and unworkable. Anytime during the dormant season where you can work the soil is fine. Plant it at a level where the top of the root ball or where the roots stop and the bark begins IS OUT OF THE SOIL. Keep soil, mulch, rocks, wraps off of the bark. This will eventually girdle a tree.
Don't stake this tree. Without leaves the wind will not push it over. It will grow a thicker trunk and better supporting roots without being staked.
Where do you live? What kind of a tree is this? Send more information and/or pictures and we can help you decide what to do. If this is a good tree, the cost to replace it is very high.
Yes, you should get rid of it.
The tree roots will go under your home and that can cause some trouble with the home's foundation and pipes as the tree gets much bigger.
But the more important reason is because if there is a storm, the wind can knock the tree over and into your home, causing some serious damage.
Like small weeds that one leaves in the ground instead of pulling, the situation with this small tree won't improve by delaying.
In general I avoid trees near the house, the well, the septic lines and near anything I don't want the roots ruining.
Pull it up... or try to relocate it... the roots might be considerable though.