If I would be accepted to UCLA ... PhD program, can I bring my Dog ... to campus?
Yes, you may bring your dog to public outdoor areas on campus, provided that you abide by various rules. From UCLA policy #135 (version at time of this post became effective in July 2019):
A Pet dog is allowed on most outdoor public areas of University Property† only if the Pet dog is under the control of a competent person and restrained by a substantial chain or leash not exceeding six feet in length. Note: Certain outdoor campus venues may not allow Pets and/or have additional restrictions.
Note that pets are not allowed on campus-provided transportation vehicles (e.g. shuttles).
UCLA's policy contains a general description of your responsibilities:
An individual who is either visiting or residing on campus and who wants to bring an animal to campus, is to be mindful that the animal may constitute a hazard to others, especially in confined or closed spaces. Such threats may include but are not limited to, allergies, exacerbation of specific phobias, falls, increased risk of disease, and bites. In addition, the presence of an animal in certain areas may distract or interrupt activities or may create a nuisance, including the nuisance caused by animal waste, noise, or damage to property.
An individual bringing an animal to campus is required to be aware of these potential threats or hazards and to exercise reasonable control over the animal in order to minimize the risk to others.
The animal owner is responsible and liable for the animal at all times while the animal is on University Property. Any injury to an individual while on University Property or damages to University Property by an animal will be the responsibility of the animal owner.
Can he come to my office ...
No, not if your office is university property†. From 135:
A Pet dog or other Pet animal (leashed or otherwise) is not permitted in any campus building ... at any time.
If your office is not university property, then the property owner's policies apply, of course.
(If you are referring to your office at your place of employment, though, fwiw bringing pets to work in office buildings is a generally unusual practice and I believe it would be very unlikely for you to find an employer that let you bring your dog to work with you, especially in a corporate setting.)
...or live with me in on-campus housing?
No (as per above), not if the housing is university property†. There are some allowances for UCLA faculty and staff in very specific conditions. See also UCLA Housing Regulations §B.11 for details specific to university housing.
There are exceptions if your dog is a service animal.
Please refer to the linked policy documents for up-to-date info, the other rules by which you must abide, and details about service animals, as this post is not a complete description of UCLA's policies, nor do I intend to update this post if their policy changes.
† Note: "University property" is defined in that policy:
University Property refers to the grounds and buildings of UCLA campus and off-campus buildings owned or operated by the University.
... (or another American university) ...
The policy I described above is UCLA-specific. For other universities, you will have to read their respective policies, which may differ from UCLA.
There isn't a central resource for university pet policies so your best bet is to visit their web site and either use their search feature if they have one available (to limit your search to official university provided info), or poke around on the site. If they do not have a way to find or navigate to policy info on their web site, then use your favorite general search engine, taking care to look at official and up-to-date sources. "University name pet policy" is a good bet for search keywords.
If you have doubts or are unable to find information, there should be some kind of contact info on the university's web site for a department that can answer your questions.
If your chosen university does not have an acceptable pet policy for on-campus housing and does not require you to live in on-campus housing, then you may also consider looking for off-campus housing that meets your needs.
Additionally, if your dog's needs are a factor in your decision, there are various city-specific resources that you can find. Lists of "top US pet-friendly cities" on the internet sometimes give rundowns of amenities available in the area (dog parks, veterinary services, etc.). You can find them by searching; I don't want to link to any specific info as it is rather subjective.