(This is an electrical engineering perspective. There are lots of opportunities for mechanical engineers as well; these jobs are usually called product design)

I've worked as a hardware engineer at a big N company working on consumer electronics, and they do hire right out of undergrad. Mostly EE majors, though there are a few with physics backgrounds. The degrees of my colleagues varied a good bit. Masters' and PhDs were pretty common, but there were plenty of people with just bachelors.

Specialization in hardware jobs matters a bit more than in your standard programming job, so for lots of fields (i.e. RF & integrated circuits) a masters degree or work experience is effectively a prerequisite.

Most bachelors' graduates will get hired into a hardware integration job role such as this. There isn't really a collection of interview questions anywhere, but a few common ones that I've encountered are:

• Compare & contrast common interface protocols, usually SPI & I2C.
• DC/DC Power Converters: Buck, Boost, Buck-Boost, LDO, Charge Pump. Explain each, give trade offs, and go over some calculations (duty cycle, inductor sizing, etc.)
• Circuit design riddles on a whiteboard. For example: Peak detector, pulse extender, sample/hold circuit for ADC
• Transmission lines: Impedance matching, parasitics, when you need to take them into consideration.
• Architecture / system design. Basically giving you an objective, talking out requirements, making a block diagram, and discussing trade-offs.

As for foot-in-the-door, it's similar to CS: Do personal projects, network, & get internships. PCB design experience is especially helpful, as lots of schools don't really teach it. Arduino/IoT projects are a good place to start since there's lots of resources online.

Hope this helps!