The Brown turkey will grow bush like if you leave it alone. I would plant the lychee on the north side of the oak if possible it will give it some frost protection,,
The brodgen likes full sun and wont need protection once it gets big enough .My Brodgen grows straight up but you can just top it. I would give it plenty of space 25' or more . Avacodos like moist well drained soil and do not like wet feet .So watering is dependent on your soil. The lychee like it a little wetter but still well drained..

Good luck

Sheeeesh- the guy we bought the avocado from said that it needed some shade because it would "burn".

So should I move it, or will the move cause more damage tree/roots?

He may have meant giving it some shade till it gets established. The Tree will do fine with some shade but full sun is needed for maximum production . Here is a U of F link

Here is a link that might be useful: Avocado Growing in the Florida Home Landscape

I agree with Red that Lychee trees (especially the newly planted and young) needs frost/freeze protection till it gets established and mature. They also need as much sun as you can give it to get the best production like most fruit trees...they also benefit from some "wind" protection when young because the very delicate new growths are very susceptible to wind damage and can stress the young tree. I built a wind break on mine and left it on for 2 years...This picture was taken 2 years ago...

Below is a great link about growing lychee trees from lycheesonline.com. Helpful information...

Here's my windbreak when I planted my Lychee tree 2 years ago. Actually the wind break served me well during a frost/freeze. Since I already had the "frame" up all I did was added a work light (for heat) underneath the tree and completely covered it with frost cloth or blanket...saved
me some time in the winter...Good luck!

Here is a link that might be useful: Lycheesonline.com

Pug, your photo made me laugh. It looks like you are growing secret plants! :)

Carol in Jax