The short answer to your title question is no. To be able to assign DOIs, one has to pay a fee to the International DOI Foundation and not all publishers are willing to pay such fees (especially diamond-model open access publishers where both reading and publishing are free and the journal is run on a volunteer basis). So, for instance, I'm pretty sure that the Electronic Journal of Combinatorics doesn't have DOIs. Most commercial and society publications do have them, and occasional low-budget ones do as well; for instance, the Journal of Graph Algorithms & Applications is a diamond-access volunteer journal but apparently Brown University was willing to cover its fees.
It's probably also the case that some journals have DOIs for some of their papers but not all of them, although I don't have examples at hand.
If you want to know whether an individual paper has a DOI, you can tell by searching the DOI foundation's guest query service. (If a paper has a formula in its title, you should probably only search for a part of the title without the formula, to avoid encoding issues.) No MathSciNet access is needed.
I've several times seen papers that do have DOIs but where the DOI was not listed in MathSciNet, so I don't think their data is complete.
