Tell us about yourself!
The wood will catch fire and burn, because the tip will reach very high temperatures
But if you use a battery it's going to discharge after some time, and it won't have any safety measures in case anything goes wrong (like your lipo battery that blew up). If you use a PSU you won't have to worry about either of these problems
What did I tell him? Anyway, you can do that, but if you have no experience it will be quite hard. You can directly connect the pushbutton to the motor (you can power it from the PSU), I would suggest to google some example circuits.
Do you mean to use another kind of power supply, like a transformer? It's certainly possible, but I haven't experimented with them.