Basement is 550 sq ft. 11x50 with a 7 ft ceiling. Am planning on using recessed LEDs since the ceiling is already completed.
11' x 50' is a rather odd shape but lighting it is done like any other space, first and foremost consider what you will be doing and where you will be doing it. Is it one big entertainment space? Is it a combination media area (tv on one side) and play area? Is part of it an office that will need task lighting? Before even considering how many or how bright the fixtures are, imagine yourself using the space.
One common rule of thumb is 1.5x the square footage in incandescent watts, i.e. if you are lighting up all 550 sq feet you will need about 825 equivalent watts, or 14 60 watt fixtures. If part of it is just a media area that won't need a great deal of light you can reduce the fixtures in that section and focus on the other areas.
Edit for future-proofing: to consider this in lumens, a living space is best lit with 20 lumens per square foot and LEDs that range from 750-850 lumens per fixture would therefore require 14-15 fixtures to fully light the area.
This would be for an even distribution of light for the space with fixtures that are around 700 lumens (the ones for calculations use 11W) - something similar to these:
At a 20 fc lighting level throughout the space (18 total for 198W):
At a 30 fc lighting level throughout the space (27 total for 297W):
Lighting isn't a set item, even these numbers shown include a lot of independent variables. The fixtures used (look for their lumen outputs) will change just how many you would need to create a sweeping, uniform light level. However if you then want some features to be illuminated for focus, such as the bar, you'd need to consider this and have higher lumen fixtures throughout that area or closer spacing of the fixtures. Also, even the finishes of the walls, floor, etc. can affect the lighting or diminish it.
20-30 fc is probably a good level to go by for general lighting that will allow you to do anything in your new room, but you can include dimmer switches if you'd also like.
Based on the comments, I've included even lower fc levels that might suit the situation a little better intentionally for just sitting around and watching tv. Again though, you'll never be able to get more than this overall level, but if you went higher you could always dim it down.
And the last point to remember here is that lighting is a gradient. Directly under the lights throughout will always be the brightest spots and this will fade out towards the walls. The reason you use additional lights is to create a uniform room wide lighting level. When you get down to the lower quantities, you'll still get an average fc level of what is shown, but it will always be darker the farther that you get from the lights (the corners and such as shown in the 5 fc range).
Hard question to answer with so little information, but I'll bite.
5x2 or 6x2 grid, spaced equally. (2 across because your ceiling is low.)
Consider splitting them into "this half" and "that half", depending on your usage.
Put them on a dimmer.
Here's a good description for full light. But, it depends what you're after. Do you want to be able to read everywhere or do you want the typical cave-like feeling (I mean "drama") of recessed lights.
28 or 22-lights (for full light) may seem like a lot & it is & it's a huge waste but aside from low profile surface fixtures I'd agree that it's your only best option for such a low ceiling. You might be able to cut the number down to 20, 18 or maybe 16, but the height of the ceiling limits the light's fan or spread pretty severely.