Actually you have the price way to low for Continual Flame.
Since it can be activated/deactivated at will and in a ring of protection the formula would be: 2 * 3 * 2000 * 2 = 24000gp. Reason for this is: 2nd level spell * 3rd level caster minimum * 2000 (use activated) * 2 not more expensive than deflection bonus of the ring but has nothing to do with it in relation therefore you double the price.
On the auto-stabilization part all you need to do is put cure minor wounds on ring, however since it needs to be trigger on something it would need Contingency to fire it off when the wearer hits negative hit points. So that brings us to the following:
Cure Minor Wounds: .5 * 1 * (2000 / 5 / 1 )*2 = 400gp. (spell level * caster level*use activated / 5 / times per day * 2 for same reason as above.) Now the tricky part here is if you want it only once a week instead of once per day you can just divide the cost by 7 and have this for about 57gp.
Contingency: 6 * 11 * (2000/5/1)*2 = 52800. Same formula as above. And again divide by 7 for once a week for about 7543gp. If you leave the stabilization part as a daily thing the price gets tricky for rings of protection +1 to +3 as the deflection bonus cost is less than the contingency cost before doubling for not having a relation to deflection. You only have this problem with a +1 ring on a weekly basis.
So you would have:
Brilliant Ring of Life Protection: This ring offers continual magical protection in the form of a deflection bonus of +1 to +5 to AC. Also upon command this ring can be made to radiate a flaming brilliance that gives light equal to a torch. Once per week if the wearer is reduced to less than 0 hit points the ring will stabilize the wearer healing one hit point in the process.
Faint abjuration & conjuration, CL 11th, Shield of Faith, caster must be of a level at least three times greater than the bonus of the ring, Cure Minor Wounds, Continual Flame, and Contingency. Price: 31,829gp (Ring +1); 39,600gp(Ring +2); 49,600gp (Ring +3); 63,600gp (Ring +4); 81,600gp (Ring +5)
Hope that helps.
RD