Let me try to answer your questions.
HDMI is "signal-compatible" with DVI, only for the digital portion, not the analog signal.
Since your original source is VGA (analog), even if you use a DVI-I (analog and digital) converter, you can't get HDMI output.
Bottom-line is, you at least need to have/derive a DVI-D source to output HDMI using a native VGA output. You can't do this by just using adaptors.
You may try to confirm this using a desktop with native DVI-I outputs. Use your DVI-I to HDMI converter and video should display correctly.
Theoretically, connecting your PC via DVI to HDMI will yield better results, as there is no conversion from one domain to another. How significant the difference is very relative and subject to one's perception.