I'm not a mechanical engineer, but I have managed mechanical engineers as part of a multi-disciplinary team in aerospace design.
My degrees are in electronics (BSc) and aerospace systems (MSc).
IMHO it all depends on where you want to go with your career and the type of organisation you end up in.
For exampe, if you want work as a project engineer (engineering management and planning), you will need to gain experience in varied engineering roles such as materials, stress engineering, verification and test, structural design, electrical/electronic design, software, safety systems, manufacturing & production, environmental etc. as well as the financial, management, legal and sales functions.
If you want to work at the cutting edge of design, then you will have to understand expicitly the capabilities and limitations of the materials, technologies, tools and techniques at your disposal. That means a much deeper understanding of the physics, equations and limitations of calculations, errors etc. in order to innovate and improve.
At the other end of the scale, if you want to work as a maintenance engineer, then you will not be taxed from a calculations viewpoint and you will be working to pre-defined plans and test schedules with computer aided everything. However, the trade off is the work will be rather repetitive and nowhere near as satisfying from my personal viewpoint.
1) K.E. Stroud. Engineering Mathematics 7th edition and Advanced Engineering Mathematics 5th edition will be invaluable.
2) Stats is used in manufacturing and production, design specification, test and failure mode analysis, modeling simulations etc.
3) In electrical and electronic engineering, pure maths is taught all the way through years 1 and 2. Complex numbers, trigonometry, matrix methods, vector differential calculus, tensors, series, domain transforms, probabiity and statistics, regression analysis, eigenvectors, Fourier analysis, Laplace transforms are covered in depth. These are applied throughout the course to all subject areas.
4) See my opening paragraphs.
5) When the going gets tough, most struggle through and see it as a a hurdle to overcome. It's only if you intend to do an M.Sc. or to go into cutting edge design work that you will need to be at the top of your game.
6) I love it. The work can be stressful with tight project deadlines, a lot of specfication and report writing, major presentations to peer groups, management, directors and clients. (You have to be prepared to justify your conclusions, methodology, costings etc. and they will and do pull it to pieces.) very long hours with social life on hold quite often when deadlines loom.
7) Career progression, I am a Principal Engineer (highest engineering grade where I work) and want to progress into senior management. Downsides, one more step away from the coal-face. Upsides, whole lifecycle management and more money. transferrable skills.