PESAs and AESAs can also do things like adopting specialized scanning patterns and pulse parameters once a contact is detected depending on the situation. They can use a different PRF and scanning pattern for tracking the detected objects while continuing to search normally, for example. Instant beam steering help with this a lot especially if a high number of objects are being tracked. If something is close they can start to use more resolution orinted parameters or decrease the resources commited on that track (less powerful pulses, less time allocated at each refresh, etc...), or a combination of both.
We know that the S1850M on the Daring can do tracking too. L-Band is not that appropriate for engagements but the said radar is a large AESA radar and the Aster series are active-radar guided munitions. Especially within 200 km of the ship, it should have no problems with guiding missiles. So as you said, the Type 45 kinda has a 4 faced AESA solution. If it faced a three vector attack that the SAMPSON can't track continuously or get overwhelmed, the S1850M too can be used for engagement. The S1850M would adopt resolution oriented parameters while contributing to the engagement. The Aster doesn't need a continuous or very high quality track either.
No. But radars are so central to warship performance that they deserve a lot of discussion. Also, examples of other solutions are useful to discuss radars. Especially since we don't have access to specific data.
My understanding is that SMART-L which the S1850M is modified from, consists only of 24 linear arrays. That's 24 linear elements, of which 16 is used for transmission. This is like an AESA type of FRESCAN array, an entire element is shaped like a blade with a wave guide that goes from end to end. This kind of array only electronically scans up and down, and is mechanically turned for the horizontal, which is about 12 RPM. This type of array is not uncommon for long range search. In terms of range, against patrol aircraft it's about 400km, which isn't any better than SPY-1 or Type 346, though ballistic targets are up to 2000km, but space has no attenuation. I am not sure and I don't think that can be used for engagement without queuing another radar. As a matter of fact, the Sachsens are upgrading by replacing their SMART-Ls with Thales NS100 or NS200 S-Band radars, reinforcing the theme that S and X bands are the best combo for warships based on design choices. L bands in general being phased out of service along with the last of retiring Ticos with the SPS-49, which is used as a secondary radar on top of the SPY-1. The question for the Type 45 is whether the pair of X-Band navigation radars it has on the main mast has any dual use capability, that can pick up low flying objects. The Burke has an S-Band navigation radar, SPS-67 (I can be wrong on the exact designation) that can also pick up low flying objects, and this being on a high mast has a better radar horizon than the SPY-1 arrays, although data will only be 2D.
The fact 054B has effectively four faces, two on the main dual face radar, another two on the secondary dual face on the rear mast. With the second being an X-Band it definitely will have fire control quality. As a replacement for the Type 364 I expect the radar to have a high rotational speed. The Type 364 has a rotational speed of 60rpm, the dual face Type 368 can have 30rpm x 2 due to the two faces.
The Type 055 definitely has eight faces if you count the secondary array set on the mast, while the 052DG has six, if you count the secondary dual face on the top mast. In addition, the 052DG has a VHF array that has a double faced 2 x 8 grid, along with an OTH X band radar and the gunnery radar. The Type 366 OTH antiship radar works by rotation, the front array being active and the second passive, and it's modified from it's Mineral-M original by allowing for the detection and tracking of low flying objects.
PESA cannot do different PRF among elements like AESA. Unlike AESA, PESA only has a single centrally controlled high powered amplifier.