Hi,
1) If you are worried about working with Jira, because you don't know the tool, don't be! The basics of Jira are really easy to understand. Even if you have never worked with it, it's almost as easy to work with as with your mail client. I'd personally disencourage you from adding a word document to the Jira Story, but using Jira's built in features. If you are lacking a field, you can always add a custom field.
If you are unsure why they use Use Cases instead of User Stories. Probably no-one outside that company will be able to truely answer that question. Every company choses their own approach or set of methods and I have never seen a client that strictly sticks to the textbook. Maybe in your case they are not an agile project and have traditionally used Use Cases. Maybe they value the more wholistic view of the Use Cases since in general Use Cases go further into details. Maybe they want the Use Case for documentation purpose.
2) If I understand your explaination well, you are talking about a software development company that works for different clients and by different clients you are talking about different legal entities and not different business units of the same corporation. Right?
If you haven't reached the critical size to seperate teams for each client, you'll have to prioritize the demands of the different clients against each other. If possible, you want to avoid working for different clients in the same sprint (set-up costs, programmers have to get their minds focused on a different technology stack every day, you constantly have to reiterate business rules of different clients, etc.). It's easier when the switch happens on a fixed date. In my experience that will give you the higher output in the long run. Sometimes you will encounter necessities that make this suggestion impossible (production defects, high-priority demands that "can't be delayed", contractual obligations considering reaction time, clients asking for demands that are too small to fill a sprint, delays from previous sprints, dependencies to external partners that don't work in your 3 weeks sprints, etc.) . Plenty of reasons why you can't keep working for one client the whole sprint.
Maybe you have been at an interview already and share your experience?