Civilization – Can a civilian unit pass through border

Seeing what units can still move

You can see whether a unit can still move from the unit list - you can open the unit list screen by clicking on the symbol and the top-left corner and selecting the 3rd option. Each unit has a colored circle next to it, as well as text explaining what it is currently doing (e.g. fortified). If there's no text, it means the unit is awaiting orders.

• If the colored circle is green, it means the unit has all its movement points remaining.
• If the colored circle is yellow, it means the unit has some of its movement points remaining.
• If the colored circle is empty, it means the unit has no more movement points remaining.

So in short, any unit you see with a green or yellow circle can still move this round, even if it's not awaiting orders - you can click it directly in that unit list screen to select it, or even double-click it to select it and center your view on it.

See all available tiles for movement

To see all the available tiles a unit can move to, either click the move-to button and then select a tile, or hold the right mouse button (don't just click it!). Then, when you release it, the unit will move to the designated spot. In both cases, you will have a blue outline showing the limit of where the unit can reach in the current turn.

If you use either of these methods and then designate a location beyond your movement range, you can even see the entire planned route (well, at least the first 8 turns of it or so), how many turns it will take to get there, and what position the unit will be in at the end of each turn.

Grey Emblem

I see you mentioned the unit emblem turning grey-ish. This only happens when a unit has no more movement points remaining in that turn. A unit may be busy doing something (e.g. fortified or constructing something) so it's not awaiting orders, but still with moves remaining, so it will still have a non-grey emblem.

No, you cannot force ranged units to perform a melee attack (but they can range-attack a unit in the adjacent hex), and this includes melee attacking to capture a city - so always bring some melee units to a siege.

They can be engaged in melee if an enemy attacks them with a melee unit, but this uses the ranged unit's strength value, instead of ranged strength, which is usually much lower - expect them to lose against unit of a similar technological level; it's best to keep them out of melee range.

The exceptions are the Gatling Gun and Machine Gun units introduced by Gods & Kings, they are explicitly designed for getting up close and personal - having only a single tile of range and matching strength and ranged strength scores that compare to the melee units of their time.

The same applies for all updated version of the archer, all updated version of the catapult, and all ranged combat ships/submarines - any 'bombarding' ranged unit of any type, essentially.
(Note: the Gods and Kings expansion divides ships in to melee and ranged, performing the same functions as their land equivalents, so this distinction is now important at sea too, especially since the melee naval units can capture now cities).

All these units can, I believe, capture civilians by entering the tile they are on - and, in the case of ships, they can destroy any embarked unit with the same method (Note: this has changed in the Gods and Kings expansion; embarked units can defend themselves and are not automatically destroyed). This always counts as the unit performing an attack, so they lose that attack until the next turn.