Personal trainer (some of the time) here so maybe I can help answer some of these.
Looking like an old and flabby harridan amongst all the strong and fit people.
This one can be difficult to navigate because we all tend to carry around the negative and vulnerable images of ourselves. I know your remark is somewhat tongue in cheek but rest assured, you won't look like an old and flabby harridan. You'll look like everyone else in the gym; a person interested in, or dedicated to, improving their fitness and strength. Most people don't pay a lot of attention to other people anyway.
Two things to remember: 1) There is usually a whole range of people working with weights, of all body shapes and sizes. 2) Even those glossy super-fit people started somewhere; some of them were once a different body shape and set a goal to change that, some of them were shy and unconfident teenagers or young people looking for a hobby or outlet. Some of them were in exactly the same position and mindset as you, just at a different time.
Besides, everyone gets that negative internal voice. Everyone. My brother's boyfriend, who is also a PT, looks like if Zac Efron and Chris Hemsworth had a lovechild. He's been on the front cover of fitness magazines. And he still has that internal voice negging on him, and has worked out strategies to overcome it.
Getting injured (what if I ruin my knees or put out my back?)
Technique and form are the key to this. Ideally, a session or two with a personal trainer with a specific goal of proper form instruction would help sort this. It's good to start with the basic compound moves - squat, bench press, deadlift, lunges, push-ups, pull-ups etc - because working out these movements (and the stabilisation process) for those will help assist most of the other movements too.
Boredom. Oh I pick something up and put it down again? Twenty times? Then another thing? For an hour?
Depends what kind of training you do but not always 20 times and not usually for an hour. If you're training for strength it's usually reps of five or six, with long rest periods in between. If you're going for muscle size it's eight to 12 reps and shorter rest periods. If you're after endurance, it's 15+ reps and shorter still rests. Some people do an hour in the gym, some do 20 minutes. You can design a programme for the time you have available, to an extent.
Weight training can also be meditative. It's oddly focused because you have to be thinking about what different parts of your body are doing, and in what order. Concentration is crucial. So although it's repetitive, in a certain way, it's also quite absorbing.
Being so weak that I can’t make any progress.
Weight training is like yoga - a journey. So don't think of it as weakness, reframe it as steps you need to take to get from A to B. Again - everyone started somewhere.