"Atomic" photon speed questions At what speed do photons travel before being emitted from an atom? In order for them to have a constant speed (i.e., the speed of light) wouldn't they have to be travelling within the atom at that speed as well, (atomic speed) else they would in fact have to accelerate once leaving the atom to attain light speed? If then they do accelerate, why should they reach a certain speed and no more?
Second question:
How many photons can an atom contain? Does it ever become depleted? 
 A: I will attempt an answer for two reasons.


*

*It may be better suited to the OP than in: What is a photon?,  

*There is an awful lot of musing and general "to and fro-ing". (if that is a word), in What is a Photon that I feel  distracts from the OP's more straightforward query.

At what speed do photons travel before being emitted from an atom.

No speed at all, because as the comment by Rococo says, photons are not like marbles in a bag waiting to be released. There are a lot of aspects of physics that we just have to accept, as we don't yet understand them, as you might see from the various viewpoints in What is a Photon.   So please accept that thinking of the electron as containing photons is wrong. If you need to think in ordinary terms, then imagine  instead that the electron can make as many photons as it needs, and they all leave at the speed of light.
The electron may be a dimensionless point, in which case it could not  "contain" any electrons. When you ask " at what speed", you are thinking in ordinary terms, but the electron is not like a soccer ball, or anything else that exists in the ordinary world. 
When two electrons are brought  close together, they emit photons that push them apart, or if a proton is found near an electron, the electron can emit photons that pull them together. So the photon can cause either  repulsion or attraction, depending on the charges involved. 

In order for them to have a constant speed (i.e., the speed of light) wouldn't they have to be travelling within the atom at that speed as well, (atomic speed) else they would in fact have to accelerate once leaving the atom to attain light speed? If then they do accelerate, why should they reach a certain speed and no more? Second question: How many photons can an atom contain? Does it ever become depleted?

Your first question assumes the photons are inside the electron, I have tried to explain that they are not.
The electron never becomes depleted of photons, because it does not have a store of them in the first place. It can go on emitting  and absorbing photons for ever. If this was not true, then we would find electrons that did not push other electrons away, and we have never found a single electron that does not follow exactly what all the other electrons do.
Try not to think of the electron in ordinary terms. It's like trying to understand  your first game of basketball, and someone has lied to you and told you the rules of soccer apply instead of the real  rules of basketball.  The quantum world is confusing to everybody and we still have a lot to understand.
