You don't need any. Your website should be hosted outside the Dominican Republic anyways for technical reasons (unless you're a bank). For more on the technical reasons please see this blog post on choosing website hosting for the DR.
Yes, you can if you live in Santo Domingo. However, even if you live in Puerto Plata, Punta Cana, Sosua, or Santiago: I can still meet you over Skype and I have many people hire me to design their website when they live far away.
Yes I can, but you should bear in mind i have to read their code. Depending on the nature of the site this can take a good bit of time or not that much. You see, with almost all of my websites (an exception being the quick and dirty Wordpress installation and configurations I do for people), are written by me from scratch. This means that I write all the code from start to finish (and sometimes I use my own templates), and therefore know it inside out.
I certainly can use the same color scheme, "feeling" and graphics/images that you used in your old site. However, I may suggest redoing it from scratch. At any rate, you can call me and talk with me about it for free.
Yes, they will. Unless we agree otherwise upfront for some reason, your site will be "mobile-friendly". It is imperative that people be able to view your site on their Android or Apple smartphone
Not in the Dominican Republic. Please see my blog post on where to host your Dominican Republic website.
Normally, yes you can. My hourly rate varies but it is usually $35.00 per hour. I live in the Dominican Republic, so I am able to charge substantially less than software engineers in the United States.
I can finish most projects within a month; sometimes significantly faster. However, this depends on what I'm doing for you. Apps require app store approvals. I may have other projects going at the same time as well. These factors can draw it out a bit. I normally specify it in the contract so we both have benchmarks.