It’s totally fine for MCMC samplers to reject draws. It’s still not good to do it, of course.
No, that’s not true to my knowledge. The attempted draw gets rejected, the sampler will “recycle” the last draw and then the sampler will try again. That’s in principle safe to do for MCMC samplers. It’s just not a good thing to happen.
… and in all honesty: When I model an ODE problem which takes long to compute and there are occasional hiccups like these, then I would try to find the root cause, but I would certainly use the model fit. I would not use that for a final decision making for the purpose the model was build for, ok… but saying that such a fit cannot be used for anything would be unreasonable in my eyes.