Property insurance: the local companies don't have a very good reputation when it comes to paying out, usually need all sorts of intrusive information, are expensive and don't offer very good cover. I use a UK-based underwriter for insuring my properties: their policies aren't cheap (but are cheaper than the BG quotes I received), the cover is very comprehensive, and they offer new-for-old cover (which the BG companies I asked for quotes had never even heard of). They also turned out to be very helpful when I had to make a claim, even reminding me that I'd omitted things that I could claim for. My premiums didn't increase the following year.
Vehicle insurance: policies are available, as in the UK, in the categories of "Third party" and "Fully comprehensive" (aka "Kasko"). Cover and premiums vary, based on the insurance agent (ie what commission they charge) plus the age and value of the vehicle. Third party cover literally that and nothing else; Kasko cover whatever risks you want covered and what percentage of the insured value you want to be covered. As an example, my wife drives a Suzuki Ignis 4x4 bought from new but now 7 years old. Third party cover is 272 leva this year; she has Kasko on top, with the insurance company valuing the vehicle at 18,000 leva and the policy covering 80% of that in case of theft/total loss - at a premium of 680 leva.
Public Health insurance: don't know, we're both covered by the NHIF (BG equivalent of the NHS)
Private health insurance: don't have any - if we need private treatment, we pay for it ourselves.
"And with health insurance, can a person emigrating legally to Bulgaria like an EU citizen or D visa holder, have just private health cover, or do they need to be paying into the public healthcare system also?"
If you're working here you're required to pay income tax, pension and Social security contributions.
"And if just a private healthcare policy is enough and that's all a person had, would they need to pay for any emergency or none emergency treatment or consultation received in a public healthcare facility if they weren't already paying into the public healthcare system?"
This explains your entitlement: https(colon)//ec(dot)europa(dot)eu/social/main(dot)jsp?catId=1103&langId=en&intPageId=4432