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Hi Jimmy, and welcome to the forum.
I'm no expert, but I can see straight away that you're flattening out your left hand under the neck to support it. Try straightening your left wrist and support the neck between the base of your index finger and your thumb - there should be space between the neck and the 'v' formed by your thumb and index finger.
You're doing quite well for a week.
As you probably noticed, your intonation is off. That's normal at this stage, and the fact that you have it down good enough to play a recognizable tune is quite good. I'd recommend playing the first 5 notes of a major scale (open, 1st finger, 2nd, 3rd, 4th) on each string to help build muscle memory, assuming you know what a major scale sounds like (and I doubt you could play that close to the right notes without it.) Chromatic scales in the same range are good, too (since you won't always be in the key that the open string starts off.)
On the bowing, the one trick you haven't got is keeping the upper arm motionless in the front-to-back plane. You want to raise the elbow up and down to change strings, but otherwise, the upper arm should move as little as possible. The theory, as I understand it, is that all the joints move the things "downstream" of them in an arc. You want to be moving the wrist, and you need to move the lower arm to get more than a few inches of bow movement. If you add in the upper arm, you have 3 arcs to manage to try and "draw" a straight line. That's more complicated than two, so it's better to keep the upper arm still.
It's possible to move the bow in a straight line with all three pieces moving (you were doing a fair job of it), but it is harder. When you get into something that requires a lot more concentration, your bowing will suffer more than if you're doing something simpler.
Related to this, but something else you're not likely to know either unless somebody tells you, is that the wrist should move up and down. At the frog (where you hold it), the wrist should be down about as far as it goes, and the bow hair should be tilted so that only the very edge is touching the strings. This is (partly) because the weight your arm exerts on the bow is much greater at the frog than at the tip.
As you move towards the tip, the wrist tilts less and less, and as you get towards the tip, starts tilting in the opposite direction (so that (if you were holding your arm in front of you) the hand is lifted up, with the knuckles higher than the wrist. This will have the bow hair flat against the strings. (Partly to compensate for the greatly lessened weight on the strings out at the tip.) (Side note: for some things, the wrist moves side to side, too. This is fairly complicated already, so I'm not going into that right now. I've been told that the motion should be like your wrist is on a string, and somebody is pulling on that string to move your wrist.)
Depending on the length of your arms and the angle you hold the violin at, you may or may not need to move the upper arm forward to get all the way to the frog. If it's just a half inch or so, I'd keep my upper arm still. There are very few cases where that last 1/2 inch is crucial. If it's several inches, then go ahead and move the arm.
The one thing my teacher has said to me more often than anything else is "Use more bow", so you want to use as much of it as you can. Practice full bow strokes a lot (all the way from frog to tip and back, at different speeds, concentrating on a good tone.) If your bow is perpendicular to the strings and moving at a decent clip, your tone will be as good as it's likely to get on that violin and those strings. (Just to be clear, I'm not saying use a full bow stroke on every note of every song. I'm saying use as much as you practically can on the notes you're playing, and practice full bow strokes as a technical exercise.)
Hope this helps,
Violin teachers usually tell you not to support the violin with the wrist, but when I started in May last year I decided to tolerate it in my case. Still I think I did the right thing: a beginner is struggling with lots of technical issues and the feeling to safely control the instrument was great help. About two months later I stopped supporting my violin with my wrist and this change was no big issue. I just came to a point where I felt safe enough to dare it.
There is something on the market which also helped me a lot. It is a product called "First Frets". It is a sticker you can fix on top of your fingerboard, with frets printed on it. This ended the frantic struggle to hit the correct positions on the fingerboard. I left my First Frets on my violin until February this year—I even performed with them three times and it gave me a comforting and relaxing feeling of safety. Without First Frets my intonation would have been very much worse. After finally removing First Frets I felt a little awkward, but after about a week I was perfectly fine. The struggle and awkwardness during my first months would have been the greater problem and I wouldn't have enjoyed playing as much as I did.
My violin is a 3/4 violin, made for right-handed players, though I play it left-handed. As I felt she was the best in the shop of all 3/4 violins I tried and the luthier agreed. I prefer Obligato strings together with Eudoxa E string. Self-made bow, weight: 24 g / 0.85 oz