For the first; Unless your bash config overrides the environment variable you've set it will use $HISTFILE from the calling shell, which is why "HISTFILE=<location>" without the export fixes it. You can test it with:
export HISTFILE=trash
bash --norc
echo hello
exit
cat trash
It isn't just intermingling either, bash may truncate the file depending on how it is configured too. You can see this by performing the same procedure as above but starting with 1000 lines in the trash file, bash will truncate the file to 500 lines by default on exit. It shouldn't happen in your case as you're also exporting HISTSIZE, but it can if you have a lot of multiline history events as bash treats them differently to zsh.
The point wasn't just about HISTFILE really though, very few of the shells own configuration variables are useful in child processes. (LC_* and PATH being the obvious exceptions that spring to mind.)
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For the second; The doc I linked to above shows how to use zkbd to handle terminal differences in a clean way. Run the wizard, and then you can use the $key array as in $key[Home]. The neat thing to do with zkbd is have the wizard run on startup when it can't find its definitions file, that way you'll get correct behaviour whenever you play with a new terminal type.
Another option is to use the terminfo database¹ directly, if you trust it to be correct for your terminals. "zmodload terminfo", and then the database is available through the $terminfo array as in $terminfo[khome] or using the echoti function².
¹ https://zsh.sourceforge.io/Doc/Release/Zsh-Modules.html#The-...
² terminfo(5) contains the mappings, but they're often far less readable than the $key array. $terminfo[kpp] vs $key[PageUp], for example.
