That's entirely up to you. There's no 'should' about it. Personally I don't want a garden full of green dumplings but some people do. I prefer shrubs to express their personalities not all look the same in slightly varying shades of green. Also, although it doesn't apply to your Euonymous, you will frequently lose blossom if you prune into a blob. The only shrub I clip into balls or cones is box which forms part of the garden's structure. But I live in a 300 year old town house with a garden not much bigger than the kitchen so I need a bit of formality.

There is a variegated boxwood with gold edged foliage. Did you get a close enough look to see the shape of the leaves? Caryopteris 'Worcester Gold' might look similar if sheared into a rounded shape. Here is a link that might be useful: link

I'd just prune out the two long branches and leave it at that. There also appears to be some all green growth to the right which needs to come off or the whole plant will gradually revert. I'd never cut a shrub with hedge trimmers unless I was into the meatball look i.e. all shrubs the same shape regardless of their natural inclinations, or obviously, I was cutting a hedge. You are right that it is crowded. Do you have another place it could go? They are easy to grow from cuttings so you could make a replacement before you move the parent so you have some insurance if the move goes wrong.

February should be a fine time to plant most cold-hardy plants. We've moved and divided mature nandinas and other plants with great success at that time of year. .... make sure that you get the (abelia) cultivar that will best fit the space. Just emphasizing this recommendation from sylviatexas. Standard abelia x grandiflora wants to grow into an 8 ft+ shrub in our area, in spite of it often being listed with a "typical" height of 4-6 ft. They also want to be nearly as wide as they are tall. They are barely recognizable in some older foundation plantings where they are sheared tight with a hedge trimmer to keep them below the windows and/or soffits. This removes most of their flowers and much of their natural character. Better to control height by annually cutting up to a third of the tallest stems, but is effort that can be avoided (minimized?) by buying one of the many smaller cultivars now available that will more naturally fit the space. Suggest shopping for the best fit (and color) for your spot(s) even if the variety is not available until later in the year; they're tough plants as long as you give them extra water while they're getting established. I wouldn't have any reservations about planting a healthy one at any time where I could get a hose to it - even in August. Our Edward Gouchers reliably max out at about 5 ft.... other than a few "cowlicks" that are snipped off after the spring growth spurt.