1. The slide/avoid contact rule - while there are a few associations that require a slide to avoid a collision, the majority of the association rulebooks (e.g. NFHS and USSSA) require the baserunner to "avoid contact." There is generally no requirement to slide.
2. The base line/path rule - this rule can be complicated for umpires, so it is not surprising that coaches have difficulty (wow, that sounded condescending. . . good). Generally the baserunner establishes his base path. The instant a play is being made on the base runner, the runner's base path becomes a direct line between his position and any base. There isn't a base path UNTIL a play is being made on the runner.
3. Appeals - a requested appeal TO THE HOME PLATE UMPIRE must be made BY THE HOME PLATE UMPIRE on a check swing called ball. Outside of this narrow rule, there is no "right" to an appeal. However, most umpires will consider a reasonable request made in a calm manner. Umpires want the call right - it is the whole reason for their presence on the field. Often once a call is made the coach is so consumed by his outrage he forgets to ask for an appeal. Especially in two man systems, an umpire will often consider a coach's request based on his partner having a better view of the play.
4. You really do get fewer calls if you are an ass and more if you're nice.
5. Obstruction / interference - again a complicated issue. Some general rules are (1) the base runner does not have an absolute right to the base path and must avoid contact; (2) the fielder cannot deny access to the base without the ball; and (3) sometimes the ball the runner and the fielder meet in the same place at the same time. We call that last situation a "train wreck" and no call is made.