IANAL, but it looks like after the fifth year after release from confinement (this includes supervision such as parole), a convicted felon may possess firearms at the premises where the person lives. There are some other caveats as well.
Here's the law:
Sec. 46.04. UNLAWFUL POSSESSION OF FIREARM. (a) A person who has been convicted of a felony commits an offense if he possesses a firearm:
(1) after conviction and before the fifth anniversary of the person's release from confinement following conviction of the felony or the person's release from supervision under community supervision, parole, or mandatory supervision, whichever date is later; or
(2) after the period described by Subdivision (1), at any location other than the premises at which the person lives.
(b) A person who has been convicted of an offense under Section 22.01, punishable as a Class A misdemeanor and involving a member of the person's family or household, commits an offense if the person possesses a firearm before the fifth anniversary of the later of:
(1) the date of the person's release from confinement following conviction of the misdemeanor; or
(2) the date of the person's release from community supervision following conviction of the misdemeanor.
(c) A person, other than a peace officer, as defined by Section 1.07, actively engaged in employment as a sworn, full-time paid employee of a state agency or political subdivision, who is subject to an order issued under Section 6.504 or Chapter 85, Family Code, under Article 17.292 or Chapter 7A, Code of Criminal Procedure, or by another jurisdiction as provided by Chapter 88, Family Code, commits an offense if the person possesses a firearm after receiving notice of the order and before expiration of the order.
(d) In this section, "family," "household," and "member of a household" have the meanings assigned by Chapter 71, Family Code.
(e) An offense under Subsection (a) is a felony of the third degree. An offense under Subsection (b) or (c) is a Class A misdemeanor.
(f) For the purposes of this section, an offense under the laws of this state, another state, or the United States is, except as provided by Subsection (g), a felony if, at the time it is committed, the offense:
(1) is designated by a law of this state as a felony;
(2) contains all the elements of an offense designated by a law of this state as a felony; or
(3) is punishable by confinement for one year or more in a penitentiary.
(g) An offense is not considered a felony for purposes of Subsection (f) if, at the time the person possesses a firearm, the offense:
(1) is not designated by a law of this state as a felony; and
(2) does not contain all the elements of any offense designated by a law of this state as a felony.
As far as the car, legal or not, I would make sure that the firearm is not within reach of the convicted felon. It is my opinion that if Jane needs to leave her gun in the car, Jane should invest in a lockable box to which only she has a key, and she should either have the gun on her person or in that box. (This is just my opinion. YMMV)
. Jane has a clear criminal record and is in no way a prohibited person (although one could argue that she should be adjudicated mentally defective for her choice of mate 
). Can Jane keep a gun in the house without fear that John will go to jail for it (assuming it is discovered)? With a gun in the house, at what point is John in possession of it? What if Jane has it in her car but John is riding with her? 
