>> ORA-01034: ORACLE not available
Database is not running. Most likely the installer did not complete all its tasks, it can happen.
For windows, the database service has to be set up (see Start/Run/services.msc), services started, installer creates the database, and quite a few other setup chores. If the database creation fails, all the other stuff is not relevant.
Under the <drive:>oraclexe folder there *should* be a datafiles (*.DBF) files somewhere. Can't recall the exact location for 10g. Try file explorer and a search for *.dbf files, that will probably come up empty.
Note the requirements, at Oracle&reg; Database Express Edition it calls for x86, not x64. The 11g XE also has lots of trouble getting a successful install on x64 as well, note the Download page states "does not work on x64" although some folks have gotten it to work.Try the deinstall steps, and use a right click/run as Administrator that *might* get past the UAC glitch that happens on the later OS versions.
A "seems to work" and actually "does work" can be two different things, especially with x64 as so many experience a failed install, enough that oracle marked the program as "does not work on x64". Since there is no Oracle Support available for XE this forum is about the only official channel to get help on problems.
XE has limitations at a total 1GiB memory footprint, 10GiB user data limit (11g), and quite a few standard and enterprise edition features are not available (see the licensing page at Oracle&reg; Database Express Edition and processor operations are limited to one CPU core, multi threading is not possible.