Flooring – Is it safe to add a floor to the garage attic

You can do a hot roof and not require any ventilation, though I would not recommend it anywhere snow accumulates on shingle roofs. Even if i don't like it, it is allowable.

Whether you need additional ventilation depends on how much you are getting now and what is allowed by the governing building code for your area. According to the International Building Code for One and Two Family Dwellings, R806 Roof Ventilation, you need the equivalent of 1/300 of the area ventilated in free vent area. If the soffit vents do not provide this, you need to install additional ventilation or opt for the hot roof concept.

Also, the attic area needs to accessible by a min 22x30 inch opening. You must also install a class 1 or II vapor retarder on the warm side of the insulation or the ventilation requirements double.

The building code does not address moisture control within the garage space, it's only concerned with moisture in concealed spaces or general ventilation of habitable spaces. You may wish to address moisture in the garage itself with exhaust fans or operable openings. This is unrelated to attic ventillation requirements.

Only the truss designer can say for sure whether they are required or not. A structural engineer might not be able to fully calculate the load transfers in the trusses, as they will only be able to estimate the joint strengths.

I'd hazard a guess that they are required or they'd not be there. From what I can tell from your photo (though it isn't quite clear), it looks like it could be one of the bracings that are required (in the UK at least) by the accepted standards for trusses (for an example, see here: NHBC pitched roof guide).

I'd be reluctant to remove it without first ensuring that you have sufficient equivalent bracing in place.