Looks like significant stem damage on the last picture. Return it and get another. Do not expect them to do any growing until it warms up in spring. Baby avocado trees won't grow at this time of year (unless you are in southern Mexico). They grow when it is warm. Mulch the area around the base with a thick (4-6") mulch, keeping the area immediately around the base of the trunk clear. The root systems are sensitive and like protection.

Looks like you have a very young tree that is responding to growing conditions in a small container; variable sun/shade, moisture, and nutrients. Avocado trees require years before fruit production begins, so you should prepare for the long term (5-7 years) with patience, and it is good you started with a grafted plant. It will have to be moved into larger containers with ample drainage as it grows; don't let it become root bound. Leaves turning brown at their tips may indicate too much water or wet feet. Also, have precautions in your Zone 7a winters, as freezing temperatures may affect your little Avocado, which would prefer at least a subtropical climate - recommended for Zones 9b-11. For more info, Google search "avocado (Persea americana)" but you won't be looking for Guacamole recipes yet. :)

It may grow past the damage. Or it may not. I personally would eat the fruit they produce--others may have a different opinion--but the badly damaged ones may not produce fruit or only produce a small amount, and it may be stunted. The brown spot on the tomato in the first picture looks like damage from rubbing against the cage. Such damage is no big deal unless a mold or something takes the opportunity to start rotting the fruit or it attracts the attention of a pest. EDIT: For future, maybe you can put your plants somewhere else on your property. If not, maybe you can talk to your neighbor and come to some kind of agreement or you can offer to pull his weeds by hand/hire a neighborhood kid to pull his weeds by hand or something, depending on what problem he's trying to solve.

I've been trying for years to help my parents in Walnut Creek, CA grow avocados bc we see and know of nearby trees, including a friend's tree that survived the all time record freeze of 1990, and they've lost several trees, always during the summer, due to these same issues of leaf browning, blackening and shriveling of stems and trunk, etc. They grow sapote, guava, and macadamia in the same part of the yard, but those darn avos won't make it. With low watering, high summer heatwaves, and clay soil, it sounds like we're getting to the bottom of things. If I can persuade them to try another Mexicola Grande, then with more water and lots of gypsum, maybe theyll have better luck. Thanks for the tips!