I used pro mix with mycorrhizae. Got it from Lowes.
I killed one of my container atemoya couple years ago from overwatering. So I am a bit timid of giving too much water. I water my container avocados twice daily cuz if I don't they would be piss (leaves curling down). But this lychee doesn't have much leaves left and I cannot tell of any indication it demands more water. Right now it's sitting under a large tree. Completely shielded from wind and direct sun lights. I have a layer of pine mulch on top and the soil feels moist underneath. I will try to increase watering daily.
I collected some rain water early in the year. They didn't last 2 weeks once the sun came out . It's a challenge to find rain water this time of the year.
I have some liquid kelp and Dyna Gro foliage fertilizer. Should I use it or wait til I see new growth? Also should it be in full shade or give it some sun light?
I would avoid using fertilizer. Due to stress, the lychee is redistributing its energy/nutrients away from outside of the canopy and inward - the roots will be unlikely to need/want additional nutrients until it knows where to send it. A 25g tree should have a very nice main trunk (and root system) so it should be able to grow back very quickly once the stresses are removed - nutrient deficiency is not an issue currently and probably could be harmful. If you do fertilize, i highly recommend flushing the soil at least once/week (add water until you see 20% of the added water flow out the bottom) to prevent harmful solutes/salts from building up in the soil that will cause fert burn. My advice would be to stop fertilizing, let the leaves come back and keep a close eye on the leaves. If young hardened leaves (<1 y old) become yellow DURING or preceding a flush, it might be signs of nutrient deficiency and you could increase the FP. Best practice for FP is probably 1 mL/gallon everyday. 2.5 mL/gallon twice weekly is also safe.
You also may want to consider that fall/winter is coming. You will get vegetative flushes every 6 weeks probably. I've had container lychees dieback multiple times (too much heat, not enough water, too cold, too hot... almost everything), and i've noticed that usually it takes 4-8 weeks for the first growth (don't touch them or spray them) which are usually somewhat disorganized and the growths are not that long. The 2nd/3rd growths also are separated by about 6 weeks and also not very impressive, but you might even see more growths from woody parts of the tree. Then the 4th/5th growth the tree really takes off with good strong growths. These obviously assume no stresses (wind, dry heat, dry soil, etc). I write this timeline because you'll notice that you'll be entering cooling weather around the 3rd/4th/5th growth, which can be bad - you don't want the tree to be accelerating growth in the fall, which is usually when the tree will start to decelerate and enter dormancy. Lychees that enter dormancy are more cold/drought tolerant in the winter.
How well protected from wind is the location? Even 15-20 mph winds can damage lychee, especially if it lost most of its leaves. I had my lychee with walls on the north & west and still got wind damage. I ended up building a wall using shade cloth and it's probably one of the best (if not the best) things i've done to help my container lychees grow. Below is a picture looking at the southwest corner and it's a Sweetheart up potted to a 45g container (too large), it lost 80% of it's leaves in late winter due to under watering and the picture is after it's 2nd vegetative flush i think. now it's growing very vigorously.
The pro mix HP myco is a peat based planting medium - it will hold a lot of water (watering every 1-2 days is probably enough; just stick your finger or a stick into the soil to see if it is moist). Peat moss does not drain very well though and will leave a perched water table (you can confirm this by tilting the container 30-60 degrees and seeing water flow out). I planted a lychee in 100% cow manure compost once - terrible idea - and had mud at the bottom but the lychee didn't mind it. Your other plants might not like it (e.g. avocados don't like wet feet from what others have said). Gardenweb's container growing forum
http://forums2.gardenweb.com/discussions/1403195/container-soils-water-movement-and-retention-xvii?n=189 has a very useful thread on water flow, and formulas for good soil.