Am I stuck with the topic/advisor that recommended my admission? I am doing a PhD in the USA. I already have a masters in France and a 5 year bachelor. I didn't think it was a bad idea to take courses again in the US, as I am changing of area. My acceptance was recomended by an advisor who is paying partially my  stipend (I receive 73% of my stipend from my government, who also covers muy tuition and partially the health insurance. The local University (not advisor) completes the health insurance cost). This year taking undergrad courses I haven't felt very challenged, and in a paper reading and discussion group I have found some professors with a more interesting (for me) approach to things.  

Am I stuck with my advisor even if usually students pick a topic after the 1st year? How can I approach this topic with whom? The graduate advisor? My PI is also the chair of the department.

Another thing is that I know some people that do not have funding at all in this department. I wouldn't want to be stuck in this situation as the rent takes around 2/3 of my income and I have my wife living with me.
This is ultimately dependent on the policies of your department. The basic issue would be that if you switch to another advisor, the other advisor would presumably be responsible for covering the remainder of your stipend. The big question is if you are free to choose both advisor and topic, or if it's expected you'd stick with your advisor, but have freedom to choose the PhD topic.

The right point of contact is the graduate advisor, who will be well versed in the procedures of your specific department.