I can't be dogmatic and say "no" outright. The TARDIS has been shown to be symbiotic with The Doctor in various ways. However, I can't agree with the reasons you give for suggesting that the TARDIS possesses the same regeneration energy as the Timelords.
Firstly, the energy that filled Rose when she looked into the heart of the TARDIS (in 'The Parting of the Ways', not 'Badwolf') actually came from the Time Vortex:
ROSE: I looked into the Tardis, and the Tardis looked into me.
DOCTOR: You looked into the Time Vortex. Rose, no one's meant to see that.
Also, The Doctor saved Rose by absorbing this energy from her:
DOCTOR: I absorbed all the energy of the Time Vortex, and no one's meant to do that. Every cell in my body's dying.
This energy is what killed The Doctor, so it is not regeneration energy. Evidently, regeneration energy is not the only golden, firey energy in the universe.
Consider though that the TARDIS itself sort of "regenerates" too. Although the change in control rooms has been explained away as simply a change of "desktop theme", in 'Eleventh Hour' the Doctor described the process of the TARDIS renewing itself as something out of his control and completely surprising:
DOCTOR: Okay, what have you got for me this time? (as he is about to enter his newly regenerated TARDIS)
We don't see any of the same golden energy emitting from the TARDIS (there is a bit of an orange glow on The Doctor's face when he first opens the door but that appears to be from the lighting and colour-scheme), although the TARDIS key does glow orange when it is ready.
I would say the implication is that "regeneration" is something of a Timelord philosophy as well as an ability. Just as they use their dimensionally-transcendental technology in things like paintings (Day of The Doctor) as well as their TARDIS, it makes sense that they would make their vehicles regenerate so they last as long as they do! Incidentally, K9 also regenerated in the Disney spin-off series, if you consider that canonical! But the ability of the TARDIS to regenerate appears to come from a different technology to their own inherent ability to physically regenerate their bodies.