I've been particularly fond of working with fountain pens lately because sometimes in the heat of the moment I press really hard and a marker tip ends up wandering into the inside of the pen... I don't use normal school fountain pen ink, but a waterproof black ink called Platinum Carbon from Japan. The Japanese also make very nice fountain pens with very fine nibs that don't clog up even with this ink - for example the Pilot Metropolitan or the Platinum Carbon. If you want a good fountain pen from Germany, google Super5. Lamy Safari with the EF nib is also always popular.
For the colored accents, I work with either the Faber Castell PITT Artist Pens with the "Brush" tip or - if I want a wider choice of colors - with the Tombow ABT finland telegram screening Dual Brush Pens. Neither is really cheap. But to start with, you don't need many colors and the pens last "forever". In both senses of the word. They contain a lot of ink and they don't dry out if you don't use them for a year or more. Unfortunately, my experience with other manufacturers has been very different. When choosing colors for sketchnotes, I like to go for pastel, even though those aren't usually my colors. They are ideal for sketchnotes because they don't stand out so brightly.
No, and I wouldn't recommend it for beginners (unless you have a lot of experience in drawing).
Sure, it looks cool when you work on a tablet (don't even try it on a smartphone!) and depending on the tool tip you use, you can get very interesting lines that are only possible with analog tools with very elastic springs or even a brush.