2. Big companies pay more, and give you access to an internal job market. Getting work is easier than you'd think, so don't worry all that much. Your first pay settlement plays a big part in your subsequent pay deals.
3. In the UK, elite graduates (oxford etc) that I work with tend to get about 1/3rd more to do the same job. They find it much easier to get onto elite graduate programmes, e.g. in finance, so I guess the employer must compete with that. STEM graduates with a 2.1 or better from a top 50 university do not have a problem finding 'graduate' work. Anything less than that, and you may as well not have a degree (i.e. you can get good jobs, but have to work your way in as you would without a degree).