Don't rely on the moisture meter to tell you when to water, at least without knowing some other things first. Moisture meters give you a result that is good in a relative sense, ie. something they say is more moist will generally be more moist, but not too useful in an absolute one, at least not before you calibrate them to some other independent indicator of moisture. So put the moisture meter aside until you are able to determine some other way how moist the soil is. I have a moisture meter and used it for a while, but these things tend to give newcomers to succulents a false sense of security.
You want the soil to go dry or almost dry at depth (not just at the surface) every time you water, ideally. There is no particular advantage to allowing the soil to remain bone dry for any extended period, but there is an increased risk of rot if you allow the soil to remain too moist too often. With a pot that big there is likely some guesswork involved. You can try poking your finger in a couple of inches if the plant has not rooted everywhere. Some people stick a bamboo chopstick in, leave it there for a bit, and see if it comes out moist or with little pieces of wet soil stuck to it. You can also try to measure how much water it takes to saturate the soil, and come to a useful result that way.
Given room temperature conditions (60sF or higher during the day), very strong light (more on this later), and reasonably good airflow, I would imagine that pot would dry out in a couple of weeks. Maybe every 3 weeks in the winter, every 7-10 days in the summer if it gets warm. Something like that. A ballpark figure for you to start with.
Water thoroughly, to saturation, until water comes out the holes at the bottom. Wait 5-10 minutes and come back for a second pass to really make sure you got the soil saturated. This is important. The cycle you need to maintain alternates between fully saturated (or close) and dry (or nearly dry). Without holes at the bottom, you have a problem, but that looks like a regular black nursery can.
You will have to wait and see with the arm that broke off a little at the base. The main thing here (with Euphorbias in general) is to avoid touching the sap, especially fresh sap, and be sure not to get it in your eyes, nose or mouth. Euphorbia sap can be a potent irritant. The actual danger seems to vary widely from plant to plant and person to person, just take a little precaution if you end up pruning or breaking off branches.
Do not repot at this point. Wait and watch and get used to the rhythm of watering in a pot that's (theoretically) filled with roots. There is much less risk of adverse consequences that way. Then when you are confident with your care, you can take the next step.
The kind of lean I'm seeing in the last picture will not be corrected by pointing the plant away from the window. I would rotate every time you water (like 180 degrees every 2 weeks, or some fraction of that if you prefer). Once the tree Euphorbias like yours get to a certain size (which yours is approaching) they start to become problematic as house plants. They need a lot of light to do well (more on this below) and especially at the growth point, where new stem is being generated. Once that point gets to the top of the window, or maybe a little short of that point, it no longer can "see" the sun, and it will start to stretch. So there's a ticking clock on indoor Euphorbia trees, but fortunately restarting them from cuttings should be pretty easy once you've got the basic care nailed down.
Provide as much natural light as you can indoors. The more, the better. Like right in front of your tallest, sunniest southerly facing window, ideally. The plant should "see" the sun for hours a day year round, ideally, with no curtain or blinds in the way.
