Of course.

So it sounds like you're currently doing it correctly.

Generally, deliverables would be the hi-fidelity mockups that the engineers would be using as the specifications to build off of. But it could also be a wireframe or even a product requirement document if there's no need for visual design work.

Deliverables are the assets the engineers will need for their future stories/tasks.

How you handle the breakdown of product stages will be based on the team and how well thought out the designs are.

Running in parallel, is when the designer or design team is doing the research, testing, prototyping, and planning for future product/feature work (backlog). So they wouldn't be part of the scrum team during that time, or whatever design resources that aren't needed for production work during that sprint cycle, would in essence be doing project work in parallel to the engineering team. The designer/design team would rejoin the scrum team when the work is at the deliverables stage.

Jeff Gothelf has a good graphic breaking down Design and Scrum: https://medium.com/swlh/here-is-how-ux-design-integrates-with-agile-and-scrum-4f3cf8c10e24

Design in scrum/agile/kanban/etc only works when the process is done in parallel to the engineering sprints, with points of intersection.

Outside of scrum: Discovery, exploration, research, iterative prototyping, journey mapping, prioritization, planning, etc.

Part of scrum: Deliverables. Ideally, 2 sprints ahead of engineering stories.

Designers who participate in the work outside of scrum, should be embedded members in the project team that will be working on that feature/product. Consistent context and deep subject knowledge are always overlooked in handoffs.

Typically, for a medium to large tech company, the process could take anywhere from three weeks to 3 months. Most companies are starting to ramp up their hiring and and onboarding plans starting in March for college hires. I've typically seen the college hire process take around 2 months. However, I would recommend that you attend college career fairs and portfolio reviews; you'll get to meet prospective employers and also get practice doing quick interviews.

It never hurts to ask about salary or welfare questions. Perhaps don't lead with those questions, but don't only ask those type of questions. Good questions to ask is about their company's facilitation of career trajectories, opportunities for further education, mentorship, etc.