| Slithery D |
I think the latter, because
The servant can perform only one activity at a time, but it repeats the same activity over and over again if told to do so as long as you remain within range.
I can see it either staying in a location and working while you're away, or following you around, but I don't think it makes sense that it's stuck in a summoning location and won't work when you're too far away.
| Crimeo |
Since it is able to manipulate and interact with solid objects, take damage from attacks, etc., so I'd say yes you should definitely be able to carry it or push it around. Things bump it, they don't pass through it, so how could it avoid being carried? It should even be able to "ride" you or your pack animals as long as no DC 11+ checks are required. (hot stuff, I know).
As for where it is anchored to, I would say you, because it does say "as long as you remain within range" but it never says anything about it being limited in range to any location, so the only text evidence seems to only be talking about the relationship between the caster and the servant.