One of the biggest points that is brought up is that there needs to be increased screening of immigrants.

Only from those who don't know what the current process is. Here is the general process. It's more complex for immigrants from specific countries (i.e. Syrian refugees go through an extensive FBI/CIA/DHS/State background check and interview process). The more prevalent version of illegal immigration is people overstaying tourist visas, which isn't prevented in any method that I've heard Trump propose (I could be wrong there).

However there are numerous instances of illegal mexicans who have committed crimes while here illegally.

Who are currently already deported... Criminal deportation is up 70% with Obama's administration.

Enforcing the current immigration laws would have prevented them.

Via a Minority Report type system of precog border guards? If we knew everyone who would ever commit a crime it'd be pretty great for the world really, not just illegal immigration.