Welcome!
1. Advice on renting a place to live for family (searched in local language but online ads are few and most are outdated; how realistic is finding accommodation below 500 Euros/month?)
Do your search in English you will find very easily the listings of the only agency specialized in long-term rentals "Rentals Montenegro" (listings in English and up-to-date). Type "long term rentals montenegro" in google and you'll find the listings. About prices, the agency made a research based on the offer it has and has come up with a grid of prices depending on the size and the quality. Have a look here.
2. Obtaining a visa or residence permit for me and my family (one EU national, and three UK, we are not married)
The most common way for foreigners to get residence permit is to first get a work permit - in your case you could set up a company and employ yourself - and then apply for residence permit (1 year to renew each year and after 5 years you can apply to obtain ID for foreigner living in Montenegro, valid 5 years). Not sure about the rest of your family?
3. Is visa required before first time trip to MN? How long does it last (how long do I have to settle ourselves)
Depending on your nationality you can first visit MN on a "tourist visa". I think for all EU and UK citizen you can stay up to 90 days. You have obligation to report your stay at the tourist office and pay the tourist tax (c. 70 or 80c per day).
4. Advice on how to employ 1-2 people for start and then hopefully expand into the local market by employing more locals (would prefer to not open up same company in MN unless necessary)
Hard to advise without knowing the field of business.
5. If required, I would love some advice on how to register a business (seen many posts with advice, would appreciate links or contacts to people who deal with this)
Usually a simple accountant can do the procedure of opening up the company for you. Some private companies offer the service, but will charge a lot for a little work. Opening a company in Montenegro is a very simple procedure which only takes a few days. Min. capital can be 1€. Monthly costs: each month you will have to pay for the accountant, and the salary of your Director (can be yourself) and social contributions. The salary (min. salary is c 190€/month) goes back in your pocket, but the accountant and contributions account for c. 250€/month.
6. Having family a car is necessity; how does it work with driving, insurance etc. in MN as EU national (UK license)
That's a bit tricky. You can import your car, but only if it is not too old (it has to respect some Euro (3?) norms. In that case you get to have local plates, and local insurance. Otherwise, you can bring an older car, and insure it in Montenegro but only for third party damage... Finally you can buy a car in Montenegro, but not in your personal name, only in your company name. In that case you can get a green card for your car from a local insurer. Most expats have their foreign cars with them. Your UK driving licence is fine to drive in MN.
7. Finally, how realistic is finding English-speaking employees in Montenegro?
Not so easy but getting better... It's a small country, so the "offer" is small! You'll find a lot of young people who studied either tourism (and usually speak English or Italian) or maritim studies.
8. How much schools cost? (private, public; are foreign children entitled to free education as well?)
In Tivat you now have 2 international schools, but they are not cheap. KSI (part of Porto Montenegro) costs around 1000€/month, and Arcadia is about half price. I believe public schools are free and accessible to foreign children.
9. Advice regards healthcare (I assume we would be entitled if I run my business there)
You get some healthcare with your social contributions, but not so sure how it works... Once in MN you'll find other expats who have more experience with this especially for the kids, and can ask them.
I hope I helped a bit!