When you convert from a 13 to a 7, the case retains the initial filing date for the 13. If the 7 is subsequently dismissed, you will need to re-file.
The answers to these questions may be different depending on your individual circumstance and should not be considered as legal advice or the establishment of an attorney-client relationship.
What you are suggesting is possible, but you may be jumping to conclusions that aren't warranted if you believe you make too much money to qualify for Chapter 7. The Median income is only a starting point for you to begin with on the Means Test. Providing you have sufficient deductions, you could still qualify for Chapter 7 but without the help of a very experienced bankruptcy attorney, you will never know for sure. Stop trying to do something that you have no experience with or be prepared to learn about the bankruptcy laws in a very expensive way. Hope this perspective helps!
You may request to have your case reconverted to Chapter 13, but make sure that you do not qualify for the Chapter 7 as one of the other attorneys has pointed out. The assistance provided to your mother may qualify as an allowable expense under Section 707(b)(2)(A)(ii)(II) - "...expenses paid by the debtor that are reasonable and necessary for the care and support of an elderly, chronically ill, or disabled household member or member of the debtor's immediate family (including parents...) and who is unable to pay for such reasonable and necessary expenses." In short, depending on how your court interprets the statute, both the house payments and the assistance to your mother may be allowable in bringing your net income to a point where you qualify for Chapter 7 relief.
It is possible to convert back or reconvert to a chapter 13. This motion would most likely need a hearing and you would need to show the court that you can fund a chapter 13 plan, but I also agree that you may want to wait and make sure that the US Trustee finds that there is a presumption of abuse in your chapter 7. They are required to file an motion and you may be able to overcome the presumption with the evidence of the financial contributions to family members and other expenses.
This answer does not create an attorney-client relationship. The information provided in this post is not legal advice but is general information. If you have questions concerning your specific situation, you should consult with an attorney in your area