When do I turn one of my pop into a specialist?
As a general answer to all your questions and as with a lot of things about civ, it's all very situational. Much of this plays into city specialization which you should read up on if you have not already.
Generally, you will want to get a couple of scientists running as soon as you get a library up in a city with enough food to support it - often your cap. Reason is to start generating GPPs toward your first Great Scientists (GS), which you can use to create an academy in your best commerce city - often your cap - or settle it likewise.
Otherwise, early game, you may start running more scientists in your cities once you reach your happy caps in which the scientist specialists will be more valuable than a food tile if that city can't grow larger. This is especially important on higher levels as much of your research will come from scientists as your economy starts to collapse due to expansion.
At what pop should I start allocating specialists?
Just gauge this according to your happy cap limitations and switch citizens around from food to specialists so that your cities do not grow until your happy cap increases. However, you do want, say, your high hammer cities to work enough mines or whatever to continue unit builds or cottage cities to work cottages so that they grow over time.
Should I have 1-3/city or 8-10/city?
There's not rule on this and it is highly situational. What you should look to do is find a good GP farm location with at least 3 food specials and a river or lake to build more farms around. The National Epic goes here and later in Caste or as you get more buildings that allow specialists, you should run as many specialists as possible. All other cities are more of a matter of what is optimal at a given point in the game. Some cities may never run specialists at all. If you are running civics like Representation, you do want to run as many specialists as possible without losing production in specials cities like your unit pumps.
Are any more useful than others?
Scientists early for sure. Libraries are generally always the first building to allow specialists anyway and academies are huge research boosts. Scientists also have huge bulbing advantages mid-game.
Late game I find Merchants are more valuable to keep gold and research up and to perform trade missions.
Engineers are nice to rush key wonders and can provide a nice hammer bonus later for building space parts (not in lieu of improved tiles, but if you have the pop to support them like with Sushi)
I use artists and prophets for golden ages or settle them. Although artists have lots of value in culture wins and prophets do have some decent bulbing options at or after Theo.
Did not follow your last question at all. However, I do suggest you check out the War Academy or sub-forum here for articles that will provide better info.
