BN3 is NOT going to be the orbital booster. It is fully stacked and is to be used for tests that do not include launches. BN3 is on a suborbital launch/test stand.
The orbital mission will use BN4 and SN20. The launch tower needs one more section to be mounted then whatever other features needed for the launch to be installed. It doesn't need the catcher feature at all at this time is first launch is not going to include landings here.
Unlike the Falcons the grid fins are not being equally spaced apart. It is thought (not confirmed by SpaceX) that this is to facilitate catching the boosters when they get to that stage.
The first of 3 vacuum Raptors needed for SN20 has been delivered for installing (ISP 378). Also 3 sea level Raptors have been delivered, are these for SN20 or BN4?
There are also hot gas thrusters delivered (needed for maneuvering the booster), some were installed on BN3 then removed as they aren't used on it as it isn't to fly. Fuel is assumed (but not specified by SpaceX) to be methane/LOX so they can use the same fuel supply as the Raptors. It is possible that this is the same design as would land the Lunar Starships on the moon but again not stated as such. ISP 300.
SN15 and SN16 are on display stands, whether they are permanent monuments or will have another fate is unclear. There has been talk of using SN16 for hyper-sonic testing.
Now will they do the orbital launch in July? Not likely (IMO). First of course they need to fully assemble and test BN4 and SN20 and that delays things from the original plan to use BN3. Second if BN3 uses 9 engines (the test stand it is on blocks the outer ring of engines so only the central 9 could fire) and those can't/won't be used on BN4 that delays things as they are currently finishing one engine every 2 days. They have recently enlarged their Raptor testing from handling 2 at a time to 5 (and have done test firings of the vacuum Raptor). So building 29 engines takes 58 days then testing (how long ??) plus potentially another 18 days (+testing) for 9 engines on BN3 so a total of 76 days, partly elapsed of course, but it could easily be September to get all the required engines. Third
this site that I've linked to before has stopped listing the orbital launch as scheduled for July to August (no precise day/time was or is listed just the month). Fourth they are having some regulatory issues over the road closures they are doing and the number of launches, potentially this is major as it COULD involve a new environmental assessment and those could take years (not happy about that part). So though SpaceX is still stating the target as being July it is highly doubtful, the regulatory problems might even block them from launching here for the rest of the year. I'd love to see where the backing for these complaints come from the locals or the competition (specifically for the Artemis Lunar Landers).
Of course Falcon 9s keep flying and landing and Starlink satellites keep going up and the BO New Glenn and its methalox engines are still no shows. Those engines are causing trouble for ULA by not appearing as promised for their new rocket causing them to have to do military launches with an older more expensive rocket and eat the extra cost. Not good for the BO/ULA National Team trying to get the HLS (lunar lander) contract that was awarded to Spacex (and stalled by objections from both these guys and the third candidate Dynetics).
The Lunar Starship apparently is not specified by the contract to be reusable. SpaceX did list it as a possibility but no promise was made in the proposal. One option if they aren't going to reuse it is to land it a second time at a planned base location to become part of the base or a fuel storage tank for when they start making fuel natively. If not used for fuel storage the tanks could be converted to rooms giving a roughly 15 story 30 foot wide tower to work with. More stories if you use lower ceilings and keep floor thickness down. Of course cutting it up for stainless steel building material would be a possible use on the base as well. Naturally you could fly it back to Earth orbit as either a fuel tank for refilling other Starships or as a Space Station (or component).