How much "catch" is adequate? Several inches (about 2 to 3 inches) is usually enough. Check compression perpendicular to the grain of the wood.
How does one slide the lagging laterally if the earth is pressing the lagging against the front flange? It is very unlikely that the earth is pressing that tightly to the back of the lagging. Although you try to not over-excavate, some always happens. The key is to make sure the gap, if any, is filled.
Should there be some blocking pounded in between the lagging ends and the webs to prevent shifting due to friction from adjusting the next piece? Most installers will install one or two large nails at each end of the lagging board and curl the nails over the front of the flanges. This keeps the boards from shifting.
...earth pressures on the back sides of the lagging installed above, in the preceding lifts, keep it in place with no mechanical assistance? Is there a practical limit here? Correct, except for the nails. For deeper holes, the wall will be braced or tiedback. This puts more normal pressure on the boards. In fact, if you do not have good contact with the earth behind the soldier beams and lagging, you can damage the beams and/or the lagging. This method of lagging has been used routinely for very deep excavations. I am not aware of any practical limit to depth.
If you know the width of the lagging boards, you can adjust the lift height to match pretty closely. Occasionally, the contractor may rip a board or install some softwood dimension lumber to fill the gap. It is standard, prudent practice to install the lagging boards with louver spaces between the boards. This louver space is most often made by nailing 2 small blocks to the top edge of each lagging board befor placing the next higher board. The louver blocks are cut from cheap, softwood 2x4's. This 1.5" gap allows you to see if there is sufficient earth behind the lagging, to add some fill if there is a void, and to stuff some hay, straw, or filter fabric if needed to control water seepage from behind the lagging. Contrary to popular reviewer comments (accusations?), louver blocks are not installed in order to save on buying lagging boards.
I was wondering why web stiffeners are not used for tieback applications. In most cases, the soldier beam webs do not need stiffeners because the web is laterally supported by the earth , lean concrete fill, lagging boards, etc. If someone designed a wall with extremely high tieback loads, possibly stiffeners could be a concern, but in that case, the soldier beams are usually much stronger also. The stiffeners you have seen may be flange stiffeners for walls with thru-beam tieback connections. In this case, there is no tieback wale and the flanges (and some times the web) are cut to allow drilling and placement of the tieback anchors through the soldier beams. This is most often used for wall-line sheeting and permanent tiedback wall with attached facings.
Installing a hole in the soldier beam usually weaken the beam. That's OK if the resulting beam is still strong enough to function. Usually, when a hole is cut, the beam is reinforced to make the beam strong enough. Reinforcement can include adding flange plates and/or adding web plates. If you are designing a wall like this but are not the tieback contractor, you can run into problems because you probably do not know the size hole the contractor will need. One way to get around this is to use double WF or channel soldier beams where the tiebacks pass between the double members. Again, the designed should know how much room the driller needs. If you pre-fab the holes, you need to be sure that the soldier beam can be installed to the proper elevation to assure the tieback anchor is where you want it. With drilled-in soldier beams this is not a proble. With driven soldier beams, you may not known how deep the beams can be driven and you may damage the thru-beam connection while driving. Even if you pre-fab the thru-beam connections, you will probably need to fabricate some extras in the field due to the need to install supplemental tiebacks for those tiebackes that do not hold their full test load.
As I said above, precast lagging should be used only in walls that can be open cut for the full height with a safe slope behind the wall. Then, the precast lagging is stacked from the bottom up with backfill being placed as the concrete lagging is set. This makes it difficult and expensive to install tieback anchors. Therefore, precast lagging panels are not a good system for tiedback walls. Weight, spacing, slope stability, tieback installation and testing, and final aesthetics are all reasons for not using precast lagging on tiedback walls or walls that you cannot build from the bottom up using a sloped open cut.