The short answer is Yes... A utility model application can be filed first, and a subsequent patent application can be filed later, claiming priority from the earlier filed utility model (within 12 months).
Sometimes, utility models are referred to as 'short-term patents', but it would be misleading just to call it a patent. You could refer to it as intellectual property.
Illegal is probably an overstatement. I don't think it would be classed as illegal, because even WIPO refers to a "short term patent". So, you could argue that it is a type of patent. However, to claim that you had simply been granted a patent would be seen as trying to claim more than you really have. Misleading and unprofessional! Maybe some advertising standards would be breached.
You should identify it as a utility model if dealing with investors, or other parties with a business interest in the intellectual property. If you're labeling or advertising a product to the general public, you can just provide the publication number; it's up to the interested party to look it up and determine for themselves what the level of protection is.
It would depend on the jurisdiction because patents are national rights as are utility models. Many countries don't have utility models and there are some which have what are called "petty patents" which are like utility models. I am only qualified to advise on UK law (and it would be a criminal offense under the UK Patent Act to mark a product so as to claim to have a UK patent if you didn't have a UK patent), but the rule of thumb for every jurisdiction must be to act honestly and reasonably. So if you have a German Utility Model then say you have a German Utility Model, if you have an Indian Patent Application, then claim you have an Indian Patent Application, etc etc
I would prefer you maintain the utility status until full patent is granted to avoid legal issues or misleading interested individual/authorities as the case may be. Congratulations and best wishes