have another coffee...
If there is no required climb gradient published on my SID departure chart, then where do I hit an obstacle? Where do the design criteria for SID's differ from OEI performance requirements?
If your charts are based on Pans Ops, the standard minimum nett climb gradient for a SID is 3.3%. I don't know about TERPs, but it is probably the same. There is no requirement by ICAO to show the SID climb gradient on the chart, if it doesn't exceed the standard gradient. I think that most countries have adopted this policy. So, to answer your first question, provided that your aircraft can sustain a nett climb gradient of at least 3.3%, you won't hit any obstacles if you follow the SID (and if it's been designed properly, of course).
The answer to your second question is a bit more complex and I think that I'd prefer to leave that one to
Old Smokey, or
John T. They have far more experience in the OEI scenarios than I do. However, what I can say is that, if your aircraft can still climb at 3.3% or better with one engine inop, you can happily and safely follow the SID.
There are 2 problems with this, however. Firstly, there are a lot of aircraft that can't achieve a 3.3% climb gradient with OEI. Second, the particular SID may require a steeper climb gradient and, if so, it will (or at least should) be highlighted on the chart. I've seen SIDs with gradients up to 4.7% and there may be some that demand even better performance than that. If you're dealing with one of those, then a 3.3% climb gradient is likely to ruin your whole day.
With those few comments, I'll pass the ball to the OEI experts!