Best way to upgrade my 2nd hand Linn system

[LTig]

[..]
To make the long story short, this are a couple of questions:
1) does it make sense to connect the preamp out of the Majik integrated amp to active loudspeakers running unbalanced? That way I could benefit of the multiple inputs of the preamp, including the phono input with no need of extra RIAA boxes.

Yes, no problem. Just make sure the speakers are sensitive enough to reach max SPL with the preamp output.

2) That given, I'd still like to directly connecting the active speakers to my EVO interface through balanced line for computer related stuff with no further passages.
Some speakers have both unbalanced and balanced inputs ( still don't know if you can keep everything connected and choose from time to time), others like Genelecs don't.
Does sort of a "switch box" exist, which could either run actives-interface-mac or actives-linn preamp-various sources without disconnecting cables every time?

Yes, even tested here, here and here at ASR.

I'd go for the balanced switch box and use a RCA-XLR adapter for the input used by the Linn. XLR/RCA switch. However if the output voltages differ a lot between Linn and EVO beware to reduce volume before switching.