Civilization – Limit civ5 AI player unit count

In the unit overview (selectable from the top-left button), you can either double-click a unit - to select it and center your view on it - or you can click a unit and then, in the lower-left unit info box, either click its portrait or its name.

As far as I know there is no way to get an immediate overview of where all your units are, which is a shame - I would also be interested in something like that. They don't even appear in the mini-map in any form until after you select them...

You can find more information about the unit list (for example, what do the colors mean) in my answer to another question.

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.