The easiest way is to shop online because you can review the different companies to get the best deals. Just make sure you know what kind of travel insurance you need. Is it for long stay or...
It depends on the company. Most are the date they receive it, Fraternals may be the date received and approved, bottom line is if you change beneficiaries with me your agent and I take the paperwork...
You can always change insurance companies in the middle of your policy date. Doing so will also result in a refund of the reminder of your policy premium since many are paid in a pro-rated way.