I have just been carving a lot of fresh beech, its my favourite to work with (apart from birch or alder)
For starters I would advise to try to not get precious with any wood, dont worry about wasting it or if you mess up when you do carvings
If you can get a chainsaw (what diameter is the trunk??) cross cut it into sections then paint the cut ends with any old gloss paint (prevents bad cracking) Its even better if you split them lenghtwise halves or quarters, then paint the ends. They will dry out, but slower and with less radial cracks. To split a log you can knock up a wood mell (maul) and 2 wedges from scrap.
I'd use some of it to make a chopping block, 2 foot length of log with 3 legs set in.....beech is ideal for that being very heavy
Beech can have a gritty feel sometimes, but it cuts beautifully with sharp tools, you can work against the grain and "come back out" of a concave cut, without a split starting (unlike eucalyptus wood LOL) Its a shame you cant get the smaller branches, they usually have zilions of decent bends and crooks going this way and that way zig zag, which are great for doing swedish style spoons
Once its dry it will take a fine burnished finish, it will take paint well, you can get sharp incised detail and crisp arrises as well, not ring porous crumbly like pine
Beech is quite stable and dense when its dried (hence why it was used for things like work benches and plane bodies) it doesnt warp badly (unlike eucalyptus wood LOL)
One thing I have been trying is carving spoons from beech, then bung them through the dish washer (60+ degreees) before they had any time to dry out....it turns the creamy white beech a lovely deep salmon pink colour, and speeds up the drying process