When the TCE directory is first created, sceboot.lst will be empty, but now you can start importing SCE extensions, run backup and have a persistent system. All your TCE stuff should now be saved on the SD card between sessions. Any files or directories you want saved that are not located in /home/tc/ or /opt/ will need to be specified in the /opt/.filetool.lst file. Just remember to run the 'backup' command before shutdown.
If you do not have any Tiny Core or dCore experience, you must read the Core book: http://tinycorelinux.net/book.html
It is also recommended you run dCore on an x86 system first to get a better understanding of operations. The Allwinner dCore port may require a refresh, hopefully the developer will have opportunity to address sometime. Everyone is a volunteer and not everyone has the hardware for testing, so it might take some time. Until then it will be difficult to provide proper support. Not to be discouraging, keep trying hopefully you'll get things running.
If your version of dCore has the readme.sh command and the Pi has an internet connection, run readme.sh and review some of the more important READMEs then the dCore wiki. The READMEs named '1st' and 'sce-import' initially most important. Later the others will help get graphics and a Window Manager running.
If 'which wget' indicates /bb/ then still the wrong version, but i'm not really sure why SSH is involved. Do you have a keyboard and monitor directly plugged into Banana Pi ? ...or is all your work through SSH from another system ?
If you have direct access to the Pi with keyboard, have a working network connection then not sure why the non-BusyBox wget issue is relevant. Without any wget changes, can you now simply run something like this successfully. Can you paste the exact error output.
When you run a wget command try specifying the wget version directly, for example:
Regarding the missing shared libraries, sorry about the extra dependencies. My dCore is running /usr/bin/wget, must have been imported from another extension, probably Dillo and running the command ldd /usr/bin/wget shows lots of dependencies. It is doubtful this wget hack will work.