Hi,
First off, in terms of which visa type to look at you would need to determine how you do on the points test. buttuming you can get a valid buttessment for your occupation and it qualifies for 60 points you would need to see if you get enough other points to get to 120. If you can't get to 120 (pbutt mark for the Send Independent 136 visa subclbutt) then you might need to look at going another route. Possible options there would be a Send Sponsored visa (subclbutt 138 or 139 - although 139 might be going away soon) which has either a lower pbutt mark (110 for 138) or no points test at all (139). With a relative in Victoria you would qualify for the 139 right now, but if that goes away a 138 might still be possible. If none of those line up then there's the regional programs (SIR and RSMS or a 137 STNI). An agent might help with this, but it's certainly possible to do yourself also.
On the medical side, it's really hard to say anything based on what you've told us. I think there's an agent (Peter Bollard?) who is mentioned here as someone who specialized is medical issues with visas, so perhaps you might want to contact him about your situation.
In terms of timing, it's going to vary substantially from case to case. My visa in total (including skills buttessment and me futzing around a bit) took just about right on a year from start to visa grant (see timeline below). Some people manage the whole thing in less than 6 months, but probably on average is somewhere between 6-12 months at the moment.
Hope that helps! Brian
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