The runner must always slide when the play is close.

There is no "must slide" rule. When the fielder has the ball in possession, the runner has two choices; slide OR attempt to get around the fielder. He may NOT deliberately or maliciously contact the fielder, but he is NOT required to slide.
Rule: 7.08(a, 3)
This rule does not apply to professionals

The runner is always safe when hit by a batted ball while touching a base.

The bases are in fair territory. A runner is out when hit by a fair batted ball, except an infield-fly.
Rules: 5.09(f), 7.08(f) Official
Baseball Rules

A runner may not steal on a foul tip.

There is nothing foul about a foul-tip. If the ball nicks the bat and goes to the catcher's glove and is caught, this is a foul-tip by definition. A foul-tip is a strike and the ball is alive. It is the same as a swing-and-miss. If the ball is not caught, it is a foul ball.
Rules: 2.00 FOUL-TIP, STRIKE
Official Baseball Rules

It is a force out when a runner is called out for not tagging up on a fly ball.

A force play is when a runner is forced to advance because the batter became a runner. When the batter is out on a caught fly, all forces are removed. An out on an a failure to tag-up, is NOT a force out. Any runs that cross the plate before this out will count.
Rules: 2.00 FORCE PLAY, 4.09
Official Baseball Rules

An appeal on a runner who missed a base cannot be a force out.

A runner must touch all the bases. If the runner misses a base to which he was forced because the batter became a runner and is put out before touching that base, the out is still a force play. If this is the third out, no runs may score. The base can be touched or the runner can be touched, either way it's a force out.
Rules: 2.00 FORCE PLAY, TAG,
7.08(e), 7.10(b) Official Baseball Rules

A runner is out if he runs out of the baseline to avoid a fielder who is fielding a batted ball.

The runner must avoid a fielder attempting to field a BATTED ball. A runner is out for running out of the baseline, only when attempting to avoid a tag.
Rules: 7.08(a), 7.09(L) Official Baseball Rules