Should I write ‘Mechanical Engineering Graduate’ or ‘Mechanical Engineering graduate’?

You should remove the spaces. Unless, of course, you are quoting a poem, in which case the slash indicates a line break:

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Wikipedia has more info:

There are usually no spaces either before or after a slash. Exceptions are in representing the start of a new line when quoting verse, or a new paragraph when quoting prose. The Chicago Manual of Style (at 6.112) also allows spaces when either of the separated items is a compound that itself includes a space: Our New Zealand / Western Australia trip. (Compare use of an en dash used to separate such compounds.) The Canadian Style: A Guide to Writing and Editing prescribes "No space before or after an oblique when used between individual words, letters or symbols; one space before and after the oblique when used between longer groups which contain internal spacing", giving the examples "n/a" and "Language and Society / Langue et société".

It takes hyphens:

all-in-one: adjective [ attrib. ]

combining two or more items or functions in a single unit: an all-in-one shampoo/conditioner.

(from the New Oxford American Dictionary)

No capitalization is required, unless it's at the beginning of a sentence, of course.