Sure you have to, however it can happen that alternate is not the choice that you can take or will take. That is why operators set their own rules as well which can be more strict than official published ones. Just to be on safe side.

as others said, takeoff minimums vary from airline to airline, however generally, if it isn't otherwise stated by the airport chart, the ICAO standard is usually 1/2 mile visibilty for 1-2 engined aircraft, and 1/4 mile visibility for 3-4 engined aircraft.

Take-off minima can be typically found in charts. I do not think airplane has any bearing on then, as typically no equipment is used during take-off. How HUGS enters into the equation, I know not.

you can set the hugs to show u the centreline when taking off to guide u

Yep I am aware of that, but since I have no experience regarding actual HUGS operation, I can not comment on how it affects actual minima. When I fly, I stick to minima given in my jeps or LIDOs, HUGS or no HUGS.Just my SOP...