While I've started a few horses that were extremely wary & much slower to come round than most, as this girl sounds, can't give you a 'timeline', because I've just taken the time it takes. I don't believe pushing or trying to force things is helpful with a fearful horse - that just tends to set you further back IME.
I think working in a smallish(not so small, that they feel trapped, & that size depends on the horse & her 'safety zone') yard can be helpful, but not big on 'round penning' as people tend to do it these days. I think it's too confrontational, at least until the horse has learned to trust you. It sounds definitely the case with this girl.
Sounds like you've been doing ok since the lasso/round penning experiences. I'd be using 'approach & retreat' methods, along with lots of positive reinforcement - that you've already been doing with treats - to VERY gradually push her comfort zones & get her confident with new things. Eg. It's unclear as to whether she's comfortable having you inside the yard when you've been feeding her, but if not, I'd start out standing in the gateway & staying there(not turned away but with relaxed bodylanguage & focus) until she settles. The instant she calms - even if it's only momentary & by degrees the first few times, turn around & leave - negatively reinforcing her behaviour. Next step when she's reliable & comfortable with that could be going in & walking around the yard - not at her, I'd start walking away from her.
Re haltering, I don't know how you'd go about sedation, short of knocking her out with a dart gun, if she's not halter broke at least. As she tolerates your touch, I'd get her better at that first, then do the same with a halter/rope in your hand. I do tend to use a noose rather than halter when I first attach a rope, but you will have to consider whether her trauma of being lassoed will make that too stressful. Either way, I wouldn't attempt to attach anything firmly until she's comfortable with it touching everywhere & draped over her neck & head, swinging it, trailing it, etc.
When first roped, I like to start in a smallish yard(big enough for them to move around/away if they feel the need) & have a long rope attached, so that I don't lose the end, but can apply pressure on it if/when I feel, because the rail will stop them finding the end of it. Again, baby steps & lots of reinforcement with teaching her to first accept & then respond to light pressure.
I suggest you look into 'clicker training', as it sounds like it's your sort of approach anyway & the principles behind it will help you understand how to progress.
Re getting her feet done or other necessary care, at this point, I would consider tranquilising her completely a better option than trying to force her into anything. I wouldn't leave her with a short rope attached to her, as for one, if she's not yet learned to yield to the halter, she's likely just to pull away, reinforce her fear & learn how to escape if you put any pressure on it.