Are there any restrictions or limitations for a gentile owning and operating a kosher establishment? Are there any limitations or restrictions for Jews patronizing such an establishment?
In modern times in the first world, if a gentile wants observant Jews to frequent his establishment, he should obtain commercial certification of kashrus (kosher status). The certifying agency will explain all he needs to do to satisfy their requirements, which will vary from agency to agency. And the only restriction on Jews will be to ensure that there is an agency they trust certifying the establishment as kosher.
Let's assume here that we're starting with all-kosher ingredients, e.g. plain produce without infestation concerns; ingredients with a kosher symbol on them; kosher meat that came certified, soaked, salted, and ready to use.
If a non-Jew wants to open, say, a grocery store where all products go straight from case to shelf, and only stock products with kosher certification, he's welcome to do so; though customers would be wise to double-check the labels. Similarly, most soda syrups and Slurpee-type bases are already kosher-certified; plenty of non-Jewish-owned convenience stores will gladly show an adult customer the packages of syrup or base that were used, so s/he can check for the certification.
Beyond that, let's break this question into three parts:
There are matters of kosher supervision in which a Jew who keeps Shabbat and kosher is trusted more than a non-Jew (or a Jew who doesn't keep those). Kosher meat (including poultry) in particular (considering how easily and cheaply it could be swapped for non-kosher meat) requires a kosher-keeping Jew to keep an eye on it on a quite regular basis. Hence, if you're a non-Jew and would like to open a kosher restaurant that serves meat, you will need a full-time, kosher-keeping Jew (known as a mashgiach, or supervisor) to keep an eye on things. Different communities will have different standards regarding how much other work he can be expected to do as well. If you want to be open on Saturdays, well, good luck finding an observant Jew to work for you those days. Practically speaking, a kosher meat restaurant is closed for Shabbat. (Yes, hotels, catering halls, nursing homes, and other places can and do make special arrangements otherwise.)
If a meat restaurant is owned by a kosher-keeping Jew, as a matter of good safeguards it's generally recommended that someone other than the owner serve as kosher supervisor.
Non-meat items (let's call it "dairy" for convenience) aren't subject to the same supervision requirements; as long as there are occasional checks, and it's overwhelmingly against the business' interests to introduce non-kosher ingredients, you're good to go. Hence there are plenty of kosher doughnut shops that are owned by non-Jews; they can be open on Saturdays too. (Whereas if a Jew owns the shop, he should be closing his doors on Saturdays.)
Cooking acts that must be performed by a Jew
Recognizing food's role with regards to culture, the Talmud mandated certain Jewish involvement in the preparation of kosher food. For bread and baked goods, a Jew should turn on the oven (or at least throw a wood chip into the fire; some commercial bakeries now have a small glow bar that the Jew turns on, then the bakery turns on the main oven element when needed). Furthermore, if you're cooking any non-baked-good food that is good enough to be served at a state dinner (so not canned sardines, but yes an entree at a nice restaurant), you must have a Jew turn on the flame first for it to be kosher -- and according to some, the Jew must also put the pot on the flame. Hence, if you're not Jewish and want to open a kosher restaurant, beyond all supervision requirements, you'll need a Jew around to turn stuff on in the mornings (and for some, to put pots on the flame all day). Again this isn't a concern for doughnut shops, as (sorry Homer) doughnuts aren't state-dinner-worthy.
Jews should give their business to other Jews, ceteris paribus
Based on a lecture I heard from Rabbi Breitowitz: All else being equal, Jews are supposed to support other Jews by doing business with them. So if I can buy the identical product from a Jew or a non-Jew for the same price, I should go to the Jew (and even be willing to pay "a little" more). Now kosher-supervision agencies will gladly supervise both Jewish-owned and non-Jewish owned restaurants, per the conditions above. So if I'm now debating between eating at Heimowitz's Pizza and O'Sullivan's Pizza, both of which have the same kosher certification, Heimowitz gets a certain preference. However, that's only if they sell an identical product. If I think O'Sullivan's is a lot tastier, I'm not obligated to buy less-tasty pizza (life's too short for less-tasty pizza). Though if there's the danger of Heimowitz going out of business, "it would be a great mitzva" to patronize him, but not required. (Though if his pizza's truly awful, maybe the biggest mitzva would be to advise him to try something else ...)
If the restaurant is gentile-owned but employs many Jews ... that gets tricky as well. The subject was actually addressed in a responsum -- To'afot Re'em OC22 about a kosher-for-Passover liquor factory that is owned by a non-Jew but employs many Jews. (Rabbi Taubes felt that giving business to a Jewish-employing company is just as good as giving it to a Jewish-owned company; "how can you tell whose Jews' money is dearer?")
The answer depends on the facts of the situation.
If you have a grocery that resells closed products, like drinks, cookies etc I don't recall anything right now that would limit you.
If you are producing/serving the food, like in a restaurant you will need to follow the rules of Kashrut. There are many details, but the question is on top of that what difference does it make if the owner is not Jewish. You may need a Kashrut supervisor, who would have control over the food and perhaps the keys to the kitchen.
One upside of a non-Jewish owned establishment, though, is that I believe you wouldn't need to sell chametz before Pesach.