I) Is it possible to follow the left-hand path, while maintaining a faith (whatever it may be), without worshipping a demonic entity? Ex: practicing Solomonic ceremonial magic.
I wouldn't describe Solomonic magic as Left Hand Path.
You are free to believe whatever you want. The question is, what are those beliefs doing? Are they strengthening you or are they subtly undermining your own authority over your life?
Is the tone "I'm playing at being in control but I know deep down that God is pulling the strings/will make everything alright in the end" or "this is just an edgy theme in a greater song that's beyond me" or, even, "I don't really understand what I'm doing but I will be loyal to tha dARk LoRD SATIN 2 tha death!"? If so, I would say that those beliefs are in conflict with a path which calls for you to become your own god and reconfigure your experience of life.
And that is the key to anything that I would recognise to be Left Hand Path, that you are the
final authority over your own life. It's a responsibility that most people cannot handle (I'm not knocking anyone for this - see below) including many attracted to the symbols of the LHP. These will tend to treat "demons" like they used to treat God, but with extra subconscious guilt, second-guessing and desperation, all of which lead to depersonalisation, confusion and unhappiness. They'd have been far better off adopting atheism as a temporary paradigm (I say temporary because there are mysteries later, but they're not helpful at the beginning) or, better yet, getting in touch with their ancestral paganism. Those frameworks will have supported a conception of the Self which is authoritative.
II) Is working with demons for various knowledge-related reasons considered incompatible with the Judeo-Christian ways?
Yes. No real Christian theologian would approve of truck with demons. You may find accounts in the lives of saints where there is command over demons but those people were saints first.
I remember having a conversation with a friend who is Orthodox and explaining the complexities of the Enochian system to him. He looked at me amazed and said "if we want to get in touch with angels, we just talk to them". In the Christian tradition, Solomon prays to God for wisdom and it is given freely; you already have a pattern in which you are seemingly invested. If you are sincere, that pattern will bring you what you want quicker than waving seals in chalk circles and it will be far less psychologically taxing.
I hold that I see things they do not, but that doesn't mean that I do not have respect for those who are properly practicing Christianity, Buddhism or other forms of the Right Hand Path. Very often, the innocence and decency which attend proper RHP practice are beguiling and enchanting to a jaded Satanist otherwise trawling through emails from damaged teenagers asking about "tha dARk LoRD SATIN" (this really happened). Certainly, I'd rather spend my time in the company of someone who is virtuous and loving rather than yet another "magician" whose conjurations haven't brought him what he desired but only the knowledge that he's acted against a conception of divinity that once brought him joy and meaning.