Mower width is important if you plan to mow between the beds, and that's usually 22 inches for a walk behind. Most people cluster beds with narrow pathways between them, with wider corridors down one side for easy access with carts, etc.

Because the planting bed is too narrow anyway, I'd remove the 3 patio blocks at the garage center and move the new plants away from the wall a little. You can trim the stumps to within 1" of grade. Plant what you wish, but 1 gallon containers will probably work best in relatively small planting holes. You should be able to get them pretty close to the desired location, or cut through old shrub roots where needed. If you don't remove the 3 blocks, you're limited to smaller, vertical plants.

In general you can plant much more intensively in a good raised bed setup that you can in traditional row gardening. I can get a dozen corn into a 3 by 3 bed NO PROBLEM and pollination is good (someone on this forum told me one year that you can really cram corn together if need be and he was right). Eggplant I do 6 to 8 plants around the perimeter of a 2.5 ft. diameter round bed and that is about maximum for that space. Your onion spacing sounds fine, you could even snug it up a little more. Cucumbers can be crowded like all heck - I do it every year. I have some 1 by 2 trellised beds in which I plant at least 6 cucumber seedlings at each end - so a total of 12 per bed. Good luck and learn from your experiments!

Really depends on what you are trying to accomplish and how much watering you want to do. If your aim is to grow sheer quantity, then closer spacing is the way to go. Google "Square Foot Gardening" or "French Intensive Gardening" and you will have a wealth of information. However, if you are looking for fewer, larger veggies, then give them space. This also helps if you don't want to have to water them constantly. Closer spacing means more watering and more fertilizing. Give the roots room to expand, and they are alot more "self-sufficient". For instance, in the Square Foot Gardening guide, spacing is as follows (per square foot): Tomatoes: 1 / ft2 (or 12" between plants) Broccoli: 1 / ft2 (or 12" between plants) Onions: 16 / ft2 (or 3" between plants) Lettuce: 4 / ft2 (or 6" between plants) Carrots: 16 / ft2 (or 3" between plants) My spacings are as follows: Tomatoes: 48" between plants in a row, about 5' between rows Broccoli: 36" between plants, about 2.5' between rows Onions: 3" between plants, and 18" between rows Lettuce: 6" between plants, and 24" between rows Carrots: 3" between plants, and 18" between rows Large plants like tomatoes are planted down the centre of a 4' wide raised bed, with a 1' path between each 4' bed. Broccoli, kale, cabbage, cauliflower, etc. are all planted in raised rows, about 1.5' wide and with 1.5' in between each row. Small items such as spinach, lettuce, onions, carrots, etc. are planted in 4' wide raised beds, but are planted the "short way", or 4' rows laid across the bed. These short rows are planted 18-24" apart, depending on the plant. The key difference here is that with square foot gardening (or intensive) you can get alot more in the same amount of space. For instance, you get 4x as many onions in the same space with SFG. But, I will bet you that they probably 1/2 the size as the ones that are given the extra room. Additionally, with the method that I use (often termed "extensive-traditional") there is alot less irrigation required. If you have the room try both methods. I'm sure you will find the system that works best for you.