Check-raising turn with initiative

I'll take a shot at this:

It's a wide range spot so your cbet turn is pretty high, guessing ~70%. You should have an equity edge with your range unless BB flats 100% on flop.

So your balancing the 30% of hands your not cbetting with some strong or monster slow plays. Slow plays should only be undertaken when you can give a free card and not give up your equity.

Villain matters but from a GTO perspective it can't be considered. We know the odds here so your giving 4:1 on the turn if you bet. So your value range to bluffing range should be at least 20%. It can be more since there is a street to come. Guessing about 25% bluffs to value.

Now come the tricky part some of your value range your check/raising .. this should be the top part of your value range. Then the top of your check/folding range becomes a bluff. On a check/raise you will give 5:1 odds 16% of your value range should be balanced by bluffs. These seem to come from hands that don't have the odds to just call a check/call, and don't have showdown value. Maybe bottom end of a OESD, or weak flush draws 54s etc. Value seems to be sets or better.

Board considerations, you might give a free card so you should only check with hands that are very strong or would like a free card. Your hand should not have show down value if your going to turn it into a bluff. However, it should have some equity. So seems like hands you can't show down but have a good amount of equity can become c/r bluffs. Then take the top of your range that has some ability to improve on the river and c/r for value.

Finally, don't forget exploitation vs a player that is only level one thinker bet/bet/bet is preferred line with bluffs and value. You need enough aggression in that villain where he will bet his hand as both bluff and value. The slow play line is more of a exploitative play vs aggressive villains than a balance play, since it splits your range and is very tough to create a balanced range on the fly.

Using all this ..

From a GTO/balanced perspective, how much to check-raise on turn with initiative?
Guessing maybe 3 or 4% of your value range and 1% of your bluffing range.

From a practical point of view, which type of hands (both bluff and value) are most suitable?
Monsters with some redraw equity for value, and draws with no show down value for bluffs

From a practical point of view, which type of board (both bluff and value) is suitable/unsuitable?
Dry boards where free card doesn't hurt you or a board that you can improve on the river.

From an eploitative perspective, which type of opponent should we hammer less or more with this move?
Exploitative is maniac and over aggressive villains that will bet when check to with to many bluffs, calling stations that bet show down hands that don't have the equity to call river. Avoid a regular that is balanced.

Thank you for a detailed and very well thought of answer. I may have somme comments, but first I need to digest your post