1. Which of the following are from the Bible?

2. God helps those who help themselves
No.
This quote is either from Sidney Algernon or Benjamin Franklin

3. Do unto others as you would have them do unto you
Yes
From Matthew 7:12. The NIV is: So in everything, do to others what you would have them do to you, for this sums up the Law and the Prophets.

4. To thine own self be true
No.
This is from Hamlet.

5. According to Jesus, I don’t have to wash my hands before dinner
This one isn’t exactly a well known quote.
Yes!
Matthew 15: The NIV: For out of the heart come evil thoughts, murder, adultery, sexual immorality, theft, false testimony, slander. These are what make a man 'unclean'; but eating with unwashed hands does not make him 'unclean.'

6. This too shall pass
No.
Not 1 Corinthians 10:12 (as I have seen claimed)
I am not sure of the origin.

7. Am I my brother’s keeper?
Yes, but out of context.
This come from Genesis 4:9 And it came about when they were in the field, that Cain rose up against Abel his brother and killed him. Then the LORD said to Cain, "Where is Abel your brother?" And he said, "I do not know. Am I my brother's keeper?" He said, "What have you done? The voice of your brother's blood is crying to Me from the ground.“
Do you really want to quote this?

8. Never go against the family
Nope.
From The Godfather.

9. As a dog returns to its vomit, so a fool repeats his folly.
Yes!
Proverbs 26:11

10. Money is the root of all evil
Not exactly.
1 Timothy 4:10 says: For the love of money is a root of all kinds of evil. Some people, eager for money, have wandered from the faith and pierced themselves with many griefs.
Notice:
the love of money
all kinds of evil

11. The road to Hell is paved with good intentions
No.
May come from Saint Bernard of Clairvaux ( ) or John Ray (1670)

12. A quarrelsome wife is like a constant dripping on a rainy day
Yes!
Proverbs 27:15
Also
Better to live on a corner of the roof than share a house with a quarrelsome wife. Proverbs 21:9, 25:24
Better to live in a desert than with a quarrelsome and ill-tempered wife. Proverbs 21:19
To be fair:
A foolish son is his father's ruin,
and a quarrelsome wife is like a constant dripping. Proverbs 19:13

13. Cleanliness is next to Godliness
No.
Maybe from John Wesley?

14. Love they neighbor as thyself
Yes
Do not seek revenge or bear a grudge against one of your people, but love your neighbor as yourself. I am the LORD. Leviticus 19:18
Jesus replied, " 'Do not murder, do not commit adultery, do not steal, do not give false testimony, honor your father and mother,‘ and 'love your neighbor as yourself.' Matthew 19:19

15. Moderation in all things
No.
Uncertain of origin.

16. The lion shall lie down with the lamb
No.
However, Isaiah uses similar imagery:
The wolf shall dwell with the lamb, and the leopard shall lie down with the young goat, and the calf and the lion and the fattened calf together; and a little child shall lead them. Isaiah 11:6
"The wolf and the lamb shall graze together; the lion shall eat straw like the ox, and dust shall be the serpent's food. They shall not hurt or destroy in all my holy mountain," says the Lord. Isaiah 65:25

17. God loves a cheerful giver
Yes
2 Corinthians 9:7 Each man should give what he has decided in his heart to give, not reluctantly or under compulsion, for God loves a cheerful giver.

18. The Lord works in mysterious ways
No.
The sentiment is certainly true, however.
Uncertain of origin.

19. Pride comes before the fall
Sort of.
Proverbs 16:18 Pride goes before destruction,        a haughty spirit before a fall.
I will leave it to you to think about whether or not these mean the same thing

20. Spare the rod, spoil the child
Sort of.
Although not verbatim, there are verses that have the same sentiment: He who withholds his rod hates his son, But he who loves him disciplines him diligently. Proverbs 13:24
See also Proverbs 22:15; 23:13-14; 29:15

21. Where there three wise men present at Christ’s birth (or shortly thereafter)?
Unknown. There are two problems:
The Bible never mentions the number of wise men.
The Bible doesn’t say exactly when they arrived. They may have visited up to 2 years after his birth.
See Matthew 2:1-18 for what we do know.

22. Which does the Bible say?
Do not answer a fool according to his folly,        or you will be like him yourself. OR
 Answer a fool according to his folly,        or he will be wise in his own eyes.
Both!
Proverbs 26:4-5 (cited out of order)

23. Sources http://www.blueletterbible.org/faq/sayings.cfm
(Googling phrases that came to mind to find their origins. I didn’t write down all of the pages.)