Spread around as a mulch if you're happy with how decomposed it is. Best time is whenever it works for you. I think the warning for your trees is not to fill the hole with compost (as implied by comments about harming the roots.) On the ground nearby is unlikely to harm. Leave some space around the tree so compost not directly touching the tree or the root ball if a bit above ground, a foot away should be plenty safe. Once established it'll be less sensitive, but still don't pile mulch or compost immediately against the trunk.

You can add compost as a top dressing anywhere at anytime.

As for digging it in - you always have to weigh the damage to existing roots and soil structure vs. the benefit of additional soil conditioning. Obviously if this is a new bed or you are planting shrubs, that is the ideal time to work compost into the first foot of soil. You have to dig the soil anyway, so might as well take advantage put in some amendments.

If the bed has deep rooted plants in it (perennials, shrubs, root vegetables), then you can safely work the compost in the top 2 inches of the soil. (I often add some fertilizer in there at the same time.) With shallow annuals or vegetables, you are better off just top dressing the bed to leave delicate roots undisturbed.

As for your new trees, I guess it depends on how you define "compost". If it is dark brown and crumbly with no recognizable pieces of the original ingredients, then there shouldn't be a problem. By all means, mix some compost in to the soil that is going back in the hole. I could see a problem if you replaced the soil with 100% compost. But mixing it in like the soil amendment that it is should be fine.

Actually, amending individual planting holes has been a long disproved methodology. Enriching - adding any sort of amendments - to an individual planting hole can create issues with root establishment as well as the more far reaching interface issues. Water will not percolate down through the soil profile evenly. The best results will come from planting directly into the indigenous, non-amended soil.

If you need to amend using compost, do so over the largest area you can manage - the entire planting bed, not just the planting hole. Or just apply the compost as a mulch or top dressing. It will still offer the exact same benefits over time.

As GG noted, don't mix it with soils. Not only the problems she mentioned, but if there's anything not completely finished, it'll suck oxygen out of the soil. That's a bigger problem, including root "burn" from anarobic decomposition byproducts.

But topdressing? Do that any time you like. Even if the compost isn't quite done, it doesn't make any difference. If you mulch mow, you drop raw matter atop the soil every time you mow that then composts.

I make it in my own bin and sometimes it's...not quite as done as I might think. The roses don't care and do well when topdressed with it. Just avoid the stems and trunks of things by about 3" or so to avoid any decay or infection problems on the plants.

I will use the compost as top dressing. I'm very grateful for the help of everyone. I love gardening, but I'm a bit of a hack at it.