A final order of removal provides the government with the basis to physically remove you from the USA (deport you). If you were under a final order of removal, then that could happen. Since you have not been issued a notice to appear, you probably have little worry about in that respect right now (but you never know).
You are asking quite a few questions and seem to have a somewhat complex situation (but not a horrible one by any means as far as I can tell). You need to consult with an immigration attorney so you can get the advice you need and advice you can actually rely on instead of pieces of information that you cannot rely on from a forum like this.
If you cannot afford an attorney, you should contact a local area legal aid and/or similar non-profit organization providing free/low fee legal immigration services to apply for help. If you cannot get free help, I highly recommend you locate some money to at least consult with an immigration attorney (there are some who offer free or very low fee consultations) and get some answers to your questions. There is no straight answer for you right now simply because there are many other possible facts that could negatively impact your case and/or perhaps provide you with other options for accomplishing your goals. You need to know what options you have available to accomplish your goals and the risks associated so you can make an informed decision. The only way that is possible is by consulting with an immigration attorney. Good luck to you and your family.
Ordered removed means that the US government has determined you should be removed from the U.S., has (according to them) given you due process to argue your side and that now ICE should remove you from the country. Unless you are appealing, you should leave or ICE will come and physically remove you.
Your case is interesting and one that I'm currently litigating for other people; one in LA and another in Seattle. If you were ordered removed, you can reopen your case if you can prove to DHS that you were simply out of money and they agree. If they haven't ordered you removed then you can file your I-751 late. Please don't leave the U.S. without fully exploring your options.
The best thing to do is hire a competent and experienced immigration attorney. DO NOT LEAVE the U. S. until more information is available. Otherwise, you can be banned from returning for ten years!!!!
You may still take action, while in the U.S. depending upon the situation. You may even try to adjust status, but this is unclear. However, you will need to determine the best option for you at this time. Good luck.
This is general information, not legal advice, and does not create an attorney client relationship.