Interpretation, not RAW
Intangibility
The "illusory copy" of yourself created by project image is explicitly said to be intangible. While it might seem obvious that intangible things can pass through a solid wall, that is interpretation rather than RAW. In fact, it is not actually clear what "intangible" means. To be more precise, the rules do not specify how an intangible thing interacts differently with the world than a tangible one. Contrast this, for example, with "incorporeal", which has a clear and consistent effect on movement and damage.
'Intangible' is used once in the Monster Manual (for the illusory self-image of an aboleth), once in the Dungeon Master's Guide (in flavor text quoting a Richard Baker story), and thrice in the Player's Handbook (in the spell descriptions of clairvoyance, imprisonment, and project image). Nowhere is is defined in a game sense, and nowhere does it say that intangible things can move through solid objects.
So, rather than worrying too much about 'intangible', let's look at the nature of illusions, especially other 'images'. Like many illusion spells, project image contains a variation on 'Physical interaction with the [illusion] reveals it to be an illusion, because things can pass through it' in its description.
It seems both logical and reasonable that if solid things can pass through the illusion, then the illusion can pass through solid things - and yet in a strict RAW sense, nowhere does it ever say that. Further, for all the illusion spells that allow one to create movable images, nowhere does it ever explicitly say they can be moved through solid objects like walls.
Many illusion spells say that their images appear at 'a spot within range'. This is notably different from creature-summoning spells, which typically say that the creatures appear in 'unoccupied spaces that you can see'. The fact that illusory images are not restricted to spaces that are unoccupied is another clue that they can, in fact, pass through solid things. It could be argued that the ability to place an illusion on 'a spot within range' rather than 'a spot within range that you can see' implies that you could start your illusion on the far side of a wall or inside a solid object. However, the counter-argument to that are the "clear path to the target" rules, which say that even if you don't need to see the point in space you are targeting, it still cannot be behind total cover to you. Thus you could not make a silent image of an illusory ghost starting inside a wall and later have it move out - although once you have made the image, you might be able to have it move in or through.
Consider also the description of major image. There, the illusory image you create "appears at a spot that you can see within range". Once you have made it, though, you can "use your action to cause the image to move to any other spot within range" - without regard to whether or not you can see the spot. This strongly implies that it would be possible to create the image on the near side of the wall, and then have it pass through to the far side. If that is a working example of how illusions function, it seems like your projected image can be sent through walls as well.
"Your speed"
As a counterargument to the illusory nature argument, the projected image is said to be able to move at "up to twice your speed". Recall that in 5e, "your speed" is not just a number, but implies a mode as well - absent any other indicators, it is a walking speed, but it could also be a swim speed, fly speed, etc. Thus by saying that the image moves at twice your speed, it could be taken to mean that the image moves at twice your walking speed if you have only a walking speed, but that it cannot move off of the ground unless you have a fly speed and it cannot pass through walls or enter the earth unless you have a burrow speed, and then it can move at twice whatever those speeds are.
RAW, nothing says that the projected image can move through walls, and in fact no illusions are explicitly said to be able to pass through physical objects. However, both project image and other illusion spells permit the image to appear in an occupied square and permit physical objects to move through them, so a reasonable interpretation is that these kinds of illusion spells can move through walls and other solid spaces. Obviously its utility as a scout and messenger will be greatly increased if it can fly and move through walls, so this seems to be largely a DM decision about what they want the spell to permit.
If it can move through earth
Project image says that "You can see through its eyes and hear through its ears as if you were in its space." So consider what your senses would be like as you passed through solid stone or earth - certainly blinded, and deafened to all but the closest or loudest sounds. How is the image to orient itself? What will keep it from going off-track? A charitable DM might also give it your sense of balance, so that it could tell up and down, but even so, imagine yourself moving through solid rock for more than a few seconds. If the DM has rules for becoming disoriented when attempting to move in the dark, these would be appropriately employed here (personally I require Survival checks for creatures attempting to move in a specific direction when they cannot see).
If you plan on having it move through the earth, and your DM agrees that it has all your senses (not just sight and hearing), the Keen Mind Feat, allowing you to always know the direction of true north, suddenly becomes very valuable. As far as a I know, there is nothing in 5e that permits one to know their depth underground, which would also be of great utility. Previous editions granted this to dwarves (for example the Deep Dwarves of 3.5e).
