Yes, imagine a highly elliptical orbit and you've just reached the furthest point , like the roller coaster at the top of the hill, almost stationary. At that point you could pretty much boost the object in any direction you wanted, and it would then assume that orbit.

Yes. In that situation, where the perihelion is at the minimum tolerable distance from the Sun, the speed at apohelion is proportional to 1/(distance to sun) (this follows from conservation of angular momentum).