Distance is always a factor in visibility if the light is able to spread out or is passing through an obstructing medium. So is relative brightness. Your eye is sensitive in a very exponential way, everyone knows the sun is brighter than the inside of house lit by light bulbs but they typically don't realize that if you're reading a book at picnic table on a sunny day it is being hit by hundreds of times more photons then if you're reading it while sitting on your couch near a lamp. Nevertheless your theoretical digital clock tower is still transmitting the same number of photons day or night. From a practical standpoint there are several factors that would limit its ability to be read though, and the backlit one at night would be visible further off but not infinitely far away.

Well an Edersphere comes to mind. There's also no reason you couldn't build a domed colony on some methane-ridden rogue planet and torch the stuff for power, assuming you didn't have fusion. Stars and sunlight are probably necessary for complex life to develop but one doesn't actually need a star to live. I don't know why one would do this but one could drag earth out into the void, stop it spinning (easier than moving it out there) and mount a huge light on the moon, or an artificial one designed to orbit once a day up at geo-synch.

The problem with saying 'anything weird' is that weirdness is going to depend on the things used to do this. Trying to figure out the actual weirdness is really tricky, unless one of the effects grossly overrides others. You'd still have weather, if the planet wasn't spinning but was being orbited by its light source, even if that light source had minimal mass, because it would be heating the air as it turned about the planet and this should still cause a general direction to the air, I think. There will still be weather patterns of some sort though, any time something isn't subjected to constant unchanging effects it becomes dynamic. If nothing else the air would flow away from the equator during the day and back toward it at night. We have some other phys/eng guys on the site who might feel more comfortable answering this one, or over on physicsforum. Non rotating planet receiving earth level light from an artificial moon/lamp orbiting it every 24 hours would be the way to phrase it. Keep in mind for this world to be unlit the light source would have to produce no significant tidal forces itself. There are several ways that might be accomplished, but it would also make for a geologically dead world, and eventually whatever weather did exist would erode it into a flat perfect sphere. If it were essentially an unmoving sphere nested inside a bigger shell that provided light in every direction at the same time, powering up then back down again, you'd still have some weather, with the air expanding during the day and water evaporating, then contracting and raining in the evening.

Hope it helps.