Thought I'd wander back in for a bit, see if there's anything interesting going on...
How far out from the star do ships have to be to safely enter/exit the warp? What are the important factors?
Marco's got the key facts covered here.
Personally, I tend to take the approach that entering the Warp near a planet is less dangerous for the ship than it is for the planet. The way I've long viewed it is that entering or leaving the Warp allows must necessarily open a hole between reality and Warp-space for a brief time, and while this hole is only big enough to let the ship through (surrounded by the pocket reality of its Gellar Field), some of the stuff of the Warp will leak back through the hole. Raw, undifferentiated Warpstuff (I shy away from calling it energy, as that implies certain things that aren't necessarily true of the ur-substance of the Warp) entering reality is dangerous
, and local physical laws can be disrupted before the Warpstuff dissipates (well, doesn't so much dissipate as become subsumed into the underlying structure of the universe as it mimics, and then submits to, the laws of physics).
At the edge of a star system, this is less disruptive than it would be in high orbit above a planet.
Returning from the Warp has additional considerations - exiting the Warp into a gravity field is dangerous because it takes you from a place where gravity doesn't exist to a place where it does, which can cause undue stresses on the ship from the sudden transition.
- How often do Imperial starships have to refuel? How long can they operate independently?
As far as I recall, Imperial vessels use Plasma Drives, which is essentially obfuscated naming for a high-power fusion reactor - essentially a miniature controlled star in the engine room. I'd argue that the ship's reactor lasts basically as long as the ship itself does, as technology that you can't turn off without fear of never being able to turn it back on seems entirely fitting for the Imperium.
- How much space-based infrastructure does the Imperium have/need?
Loads, but it's distributed unevenly. Individual worlds tend to focus on a small number of contributions to the Imperium, rather than being self-sufficient. Consequently, one world might be devoted almost entirely to the mining and refinement of raw materials (with zero consideration of ecological impact; worlds are used up and abandoned by the Imperium), which are in turn used on other worlds to produce things. Those worlds would be fed by a mixture of local subsistence food production, and planetary-scale agriculture on other worlds.
Local subsector- and sector-based trade infrastructure is needed to move raw materials from refinery to factory, food from farm to table, arms shipment from factory to fortress world, and so forth, in a continual cycle of symbiosis. These vessels will use stable, regularly-travelled, regularly-mapped routes through the Warp, keeping travel times brief and perils few (and requiring less-skilled Navigators - better navigators are reserved for those who are likely to travel 'off the beaten path', such as the military).
The Administratum - on a Segmentum and Sector level - prioritises worlds based on their contribution to the Imperium. The more a world can give, the more the Imperium will demand in tithes. Worlds with heavy tithes will likely have a large Imperial presence (as opposed to local planetary authorities), as will worlds designated as administrative capitals, but by and large as long as a world continues to do as it's told, the Imperium doesn't care how the planet is run. The Imperium will get involved as and when something problematic occurs. The trade infrastructure (and thus spaceports and similar apparatus) of a planet will be determined almost entirely by the expected scale and form of the tithe - you have different expectations for a world producing Promethium than you do for a world which pays its tithe in soldiers.
- How does piracy work?
The overwhelming majority of piracy is in-system, as pirate vessels are seldom warp-capable, and the few that are capable of entering the Warp won't have a Navigator, so they're limited to short-range raids. This makes them relatively easy to stop, were it not for the fact that the Imperial Navy is really busy, with individual sector Battlefleets spread thin. Local PDF forces can have ships, but they can't be Warp-capable, and they're unlikely to have the firepower and raw might of the Imperial Navy. Similarly, some parts of the Imperium operate void-capable Arbites precincts to handle piracy and similar lawlessness, though this is far from universal. Most pirates probably survive for as long as they do entirely because the Imperium has better things to do - only those that become a real problem draw enough attention to make the Imperium deal with them.
Long-range piracy is a mixture of renegades/traitor vessels (Navy who have mutinied and set out on their own - happens more often than people like to admit), Chaos forces (which have infrastructure that most pirates don't, but which tend to be raiders and reavers more than full warfleets), and aliens (Orks, Eldar, and a few other minor alien races). This can be a serious problem, worthy of Navy attention, but their ability to travel long distances makes them harder to catch.