Does God ever intentionally put things in our way himself to allow us to be tested (as the Psalms says)? Or does God merely allow Satan to give us trials and temptations?
One piece of information regarding "testing". The King James version (1611) does not use the word "test". This is because the first English use of the word "test" does not occur until the late 1600s. It was first used in regard to metallurgical testing.
The concept of testing is found throughout the Bible often using the word "trial". Even the metaphor for metallurgical purification (refining) is used.
Isaiah 1:25 And I will turn my hand upon thee, and purely purge away
thy dross, and take away all thy tin:
Zechariah 13:9 And I will bring the third part through the fire, and
will refine them as silver is refined, and will try them as gold is
tried: they shall call on my name, and I will hear them: I will say,
It is my people: and they shall say, The LORD is my God.
However the concept of measurement implicit in testing is not so much a Biblical component. God already knows all about us. The testing that is done is to provide opportunities for correction, transcendence, and faithful dependence.
Hebrews 12:6 For whom the Lord loveth he chasteneth, and scourgeth
every son whom he receiveth.
The idea is to allow various difficulties such that we benefit developmentally.
James 1:2-4 My brethren, count it all joy when ye fall into divers
temptations; Knowing this, that the trying of your faith worketh
patience. But let patience have her perfect work, that ye may be
perfect and entire, wanting nothing.
Romans 5:3-5 And not only so, but we glory in tribulations also:
knowing that tribulation worketh patience; And patience, experience;
and experience, hope: And hope maketh not ashamed; because the love
of God is shed abroad in our hearts by the Holy Ghost which is given
unto us.
God will directly act to correct us. God may even directly act to provide us a circumstance for growth. God may also allow the evil of others to benefit us or others.
Genesis 50:20 But as for you, ye thought evil against me; but God
meant it unto good, to bring to pass, as it is this day, to save much
people alive.
God may even make use of an evil spirit to manipulate the unsaved.
1 Kings 22:20-22 And the LORD said, Who shall persuade Ahab, that he
may go up and fall at Ramothgilead? And one said on this manner, and
another said on that manner. And there came forth a spirit, and
stood before the LORD, and said, I will persuade him. And the LORD
said unto him, Wherewith? And he said, I will go forth, and I will be
a lying spirit in the mouth of all his prophets. And he said, Thou
shalt persuade him, and prevail also: go forth, and do so.
However, God does not try to get us to succumb to sin.
James 1:12-13 Blessed is the man that endureth temptation: for when
he is tried, he shall receive the crown of life, which the Lord hath
promised to them that love him. Let no man say when he is tempted, I
am tempted of God: for God cannot be tempted with evil, neither
tempteth he any man:
There are two ways we can respond to a trial or test. We can with the help of God overcome, or we can give in to our base desires.
James 1:14-15 But every man is tempted, when he is drawn away of his
own lust, and enticed. Then when lust hath conceived, it bringeth
forth sin: and sin, when it is finished, bringeth forth death.
God limits the severity of our temptations.
1 Corinthians 10:13 There hath no temptation taken you but such as is
common to man: but God is faithful, who will not suffer you to be
tempted above that ye are able; but will with the temptation also make
a way to escape, that ye may be able to bear it.
God understands our temptations.
Hebrews 4:15 For we have not an high priest which cannot be touched
with the feeling of our infirmities; but was in all points tempted
like as we are, yet without sin.
For those who live by faith, God may chasten, correct, or test either by his own hand or allowing circumstances to provide such situations. While frequently uncomfortable, these situations are always for our good.
Romans 8:28 And we know that all things work together for good to
them that love God, to them who are the called according to his
purpose.
The purpose of such trials is developmental.
Romans 8:29 For whom he did foreknow, he also did predestinate to be
conformed to the image of his Son, that he might be the firstborn
among many brethren.
Ephesians 4:14-15 That we henceforth be no more children, tossed to
and fro, and carried about with every wind of doctrine, by the sleight
of men, and cunning craftiness, whereby they lie in wait to deceive;
But speaking the truth in love, may grow up into him in all things,
which is the head, even Christ: