multi-wire branch circuit breaker question
You're correct that there are several problems here.

First of all, if indeed breakers #4 and #8 are sharing the same neutral wire in a 14/3 with ground cable, then you're at risk of overloading that neutral.  Correct MWBC wiring requires either a two pole breaker, or two adjacent breakers connected with a listed handle tie.  Whether AFCI is required will depend on your local code, but it isn't a bad idea in any case.  GFCI is required on the outdoor outlets, and it's not a bad idea to do that in the breaker, but using a GFCI outlet (preferably weather-resistant rated) is also fine.  

Additionally, 20A breakers are not allowed on 14 AWG wire.  When you're replacing the breakers, get a 15A.  

However, none of this should cause the outlet breaker to trip when something is plugged in (unless perhaps if it's already a GFCI breaker), so it's very possible that there's something else wrong in the wiring as well.  Since your family member made so many mistakes already, it would be wise to open up all the lights and outlet boxes they wired, and check their work (or hire an electrician to do so, if you don't feel confident doing so yourself).  

Edit: since you mentioned breaker #4 is a GFCI breaker, the fact that its sharing a neutral with breaker #8 is why it's tripping.  GFCIs function by monitoring the current on both the hot and neutral and making sure they're equal, and if not, they trip.  Since the circuit shares its neutral with breaker #8, the currents will not be equal, so the GFCI is doing its job and tripping.  To have a GFCI breaker protect a shared-neutral MWBC like you have here, it must be a two-pole GFCI breaker, so that it can monitor both hots and their shared neutral, and make sure the current on all three wires sums to 0.