If someone signs a plea agreement for 3-5 years probation for drug possession and child engagement but does not show up for sentencing what would happen?

Some people do try to escape their past and a few are successful... but in the modern age of technology, it's a little harder than it used to be.  If you do not show up, you are likely to loose any deals that you have worked out.  If you are rearrested, you could face more time than you originally bargained or hoped to get.

You have a couple of additionally questions which I'll take up in order:

First, the lengths they will go to find the person will depend on the agency that is involved in your charges.  Some will simply enter a warrant and wait for it to show up when you go to renew your license or you get pulled over on a traffic stop.  Other agencies can get more aggressive and do extensive searches that involve contacting all of your known relatives to find you. 

Second, if a person leaves and stays out of trouble, it could help to reduce the negative effects, but it's still a risky move-- because many judges do not like defendant's that intentionally abscond. 

Third, many people do escape their pasts, but it's getting harder.  More and more states are getting better at their reporting requirements.  Warrants now travel electronically and can be found via a simple criminal background check in virtually any state.  You're only real hope of it not popping up is if someone forgot to enter the warrant in the database.  Warrants usually have a way of popping up at the absolute worse time.  Even though "escaping" your past is doable, it will involve looking over your shoulder for a number of years.  It will be much easier in the long run to just do your probation now and get the past really behind you-- so that it won't haunt you later.