To convert backdoor draws into probabilities, you need to multiply the probability of hitting the first card by the probability of hitting the second card given that you hit the first card.
How do I measure the probability that my straight backdoor draw will be realized at the river?
It depends on the number of gaps you have in your draw;
Suppose we modifiy your example so that your straight draw has 2 gaps:
Hole cards: 2:diamonds:3:diamonds:. Flop: T:hearts:6:diamonds:J:spades:.
The only combination of ranks that will give you a straight is 4-5. Then,
the first card must be one of the 8 cards of these ranks (4c, 4d, 4h, 4s, 5c, 5d, 5h, 5s).
the second card must be one of the 4 cards of the other rank (if, for example, a 5 was dealt on the turn, the 4 cards are 4c, 4d, 4h, 4s).
Probability of hitting the first card: 8/47; Probability of hitting the second card: 4/46;
By multiplying the two (because you need the first AND the second card), you get (8/47 * 4/46) =~ 1.48%
With one gap in your straight draw. For example:
Hole cards: 2:diamonds:3:diamonds:. Flop: T:hearts:5:diamonds:J:spades:.
Now, there are 2 types of combinations that will give you a straight: A-4 and 4-6.
Each of them has 1.48% probability, so you have 2.96% chance of hitting.
And with no gap, like in the example you gave, there are 3 types of combinations that will give you a straight: 2-3, 3-7 and 7-8. Each of them has 1.48% so you have 4.44% chance of hitting your straight.
The probability of making three of a kind ("trips") of 5 or 6?
What about two pairs?
The easiest way is to combine these two outcomes.
Not considering the turns and rivers with ranks matching the cards of the board, you will hit a trips or two pair if:
the first card is one of the 6 cards matching the rank of your hole cards (5c, 5h, 5s, 6c, 6d, 6s).
idem for the second card (but there are only 5 of them left in the deck).
Probability of hitting the first card: 6/47; Probability of hitting the second card: 5/46;
By multiplying the two, you get (6/47 * 5/46) =~ 1.4%
By the way, the same method will give you the probability of hitting a flush:
Probability of hitting the first card: 10/47; Probability of hitting the second card: 9/46;
By multiplying the two, you get (10/47 * 9/46) =~ 4.16%
Randomly dealing a turn and a river on the example hole cards and flop from your example 10 000 times got me these results:
Two Pair: 99
Straight Flush: 21
Two Pair with board cards: 691
Trips with board cards: 81
Other: (No Pair, One Pair): 8228