In case it isn't clear, Avios are the currency used to get free* flights and upgrades; tier points are used only to move up levels.

With that in mind, First on British Airways gets +100% Avios compared to economy but +500% tier points**; Club World/Club Europe (Business class) normally get +50% Avios and +300% tier points**†. However, flying First on another carrier is only +50% Avios, and Business on another carrier is +25% Avios. However, one gets the same tier points on other oneworld carriers as on BA.

For some examples, consider a flight from LHR-ORD, which can be BA or AA.

• whY bookings get 3950 Avios and 35 TPs on either carrier
• High economy buckets (YBH) still get 3950 Avios but get 70 TPs (+100%), again whichever carrier you use.
• Mentioning for completeness, BA's World Traveller Plus (premium economy) would be 4938 Avios (+25%) and 90 tier points (+150%, rounded)
• Business class gives 140 tier points (+300%). On BA, the Avios would be 5925 (+50%). On AA, it's 4938 (+25%).
• First class is 210 tier points (+500%). On BA, 7900 Avios (+100%), but flying on AA you get 5925 Avios (+50%).

Silver and Gold members also get an additional +100% Avios if on a BA-marketed flight, but the same tier points.
All four levels, yes. BA flights in JCDRIWETYBH classes may be upgraded using Avios; IB and AA flights need to be in JCDYB. Upgrades can only be by a single class; from economy to business (possible only if the aircraft doesn't have premium economy) or business to first is 100% the base cost (the cost of booking the flight outright in economy), economy to premium economy or premium economy to business is 50% the base cost.

*NB flights are not really free due to YQ etc.
**Y, B, and H booking classes grant 2× the usual economy tier points.
†Formula is different on some long routes within Europe.