This depends on the individual insurance. I would call your spouse's insurance. I wouldn't say anything about why you lost your COBRA unless they ask especially if you are well now and expect to stay that way. In other words, you don't have a chronic condition. The less you say to health questions, the greater your chance for coverage. Please don't be deceitful, but don't think you're being dishonest by not offering information. The insurance company will ask the information necessary to ensure they will make a profit from the deal if they can insure you. Good luck.
Also, if you can't be picked up by a spouse's insurance, some states (such as Tennessee) offer insurance to the uninsurable. While this may not be the best insurance, it is definitely better than none.
You should be able to. Typically, on a company sponsored policy there is an "open enrollment" period to start, end or change benefits. Most companies allow employees to make changes for "major life events" which generally include births, deaths, marriage, divorce or change in spousal benefits.
Probably. It would be a "qualifying event" and therefore you shouldn't have to wait for open enrollment.