Laundry drain pipes are required to be 2", and you are required to drain the laundry through the use of a standpipe or another indirect method like pumping into a properly plumbed laundry tub. Because of the height requirements of a laundry standpipe, the best location for it is usually directly behind the laundry equipment. A properly sized laundry standpipe would be tall enough that it is usually visible behind the washer. This is dictated by local as well as national codes. The separate standpipe is required. You can't just join a drain line to the main drain. It must be a standpipe with the P-trap in the wall. THis protects you against sewer gas escaping into you home and potentially killing you. So, while a licensed plumber can possibly move your laundry standpipe to a more convenient location behind your washer for you, he will have to remove the drywall in that section to do so, and the standpipe must remain at the current height. A licensed plumber who will pull a permit for the work and have it inspected is your best choice. Flooding your home through an incorrectly installed laundry drain is a very real possibility, and your insurance would probably not pay if the install was not up to code.

You are absolutely correct, Generally the excavation is the difficult & expensive part of your repair, but you seem to have that under control. Now as we discuss the plumbing part of the job allow me to give you the correct trade nomenclature so you will be on the same page with your plumbing suppliers and inspector. Technically speaking, what you are working on is not a drain line, it is the "House Sewer" line. Code requires a main cleanout within 3ft of the point where the house sewer enters the structure. In the case of slab construction it would be found outside and approximately 3' from the foundation. For houses with a basement the main cleanout is generally just inside the footer wall and for houses with a crawlspace it can be inside or outside the footer wall. The line from the main cleanout to the septic tank or municipal main is defined as the house sewer while the main line inside the house is defined as the "House Main Drain". This is important to know because the codes allow different materials for each type of line. The raw cut end of a pipe that has no threads or fittings is called a "Spigot end" or simply a "Spigot" The formed female end of your pipe is called an "End Bell" and pipe which has an end bell on one end and a spigot end on the other end is called "Hub & Spigot" pipe so that glazed clay pipe that you have is "Vitrified clay hub & spigot pipe". It was generally made in 4' sections and when laid, the hub is always placed on the upstream end. While it can be done, as a general rule it is nearly impossible to cut that type of pipe without breaking it. (I have made & attached an illustration to help you understand the text.) To begin your project excavate until you find the end bell on a section of pipe downstream from where you want to install your cleanout. Fernco makes a rubber donute gasket that will fit inside the hub of that clay pipe, then your PVC will be inserted into the center of the Fernco. (make sure the Fernco you buy is listed as "Vitrified clay x PVC") If you can not find the correct Fernco some jurisdictions will allow you to pack the joint with "Oakum" which is a fiberous rope looking material that is impregnated with tar. To install the oakum you wrap it around the pipe and push it into the joint with a broad screwdriver, packing as tight as you can. You can make your cleanout with a Wye & 1/8th bend (sanitary 45deg elbow) in the manner you describe, but what you are making is a running cleanout and in some jurisdictions they require a running cleanout to be installed to the line can be snaked in both directions. The preferred fitting is a "Rodding Tee" sometimes called a "Cleanout Tee". An ordinary sanitary tee has a radius towards the downstream direction of flow, but a Rodding Tee has a radius in both directions. In some jurisdictions if they require a bi-directional cleanout and if you use the Wye & 1/8th bend method you actually have to make two cleanouts, one in each direction. Also not that the one for snaking in the upstream direction must be installed on the downstream side of the downstream cleanout. That is to insure that you can snake the complete line. If they were reversed you could not snake the space between the two cleanouts. You then continue your new PVC pipe to the Spigot end of the vitrified clay pipe on the upstream side of your repair and connect the PVC to the clay pipe with a Fernco coupling. (Careful here. Although both the clay pipe & the PVC are 4", that is the ID of the pipes, but because the clay pipe wall is much thicker than the PVC pipe a common 4" Fernco won't fit. Make sure you get a Fernco coupling that is listed as 4"PVC x 4" Vitrified Clay. (Although I have on occassion found them in Home Depot & Lowes as a rule you will probably have to go to a real Plumbing supply house to find the correct Fernco's) Now to answer your next question, how to cut the PVC. You can easily cut PVC with a hacksaw, a PVC saw or what I prefer for cutting 3" and larger, a common hand wood saw. Here is a tip when cutting pipe with a saw. After you start your cut, when you are about 1/4 of the way through the pipe, rotate the pipe a 1/4 turn, cut down another 1/4, then rotate the pipe again and finish your cut. Rotating the pipe in that manner will help keep a straight cut line on the end of your pipe. Sand is the preferred material for bedding the pipe and continue covering with sand until you have a minimum of 4" on top of the pipe, then you may use "Clean fill dirt" that is free of rocks until you have 12" above the pipe. When you finish backfilling it is best to have the top about 3" or 4" above grade to allow for settling. One a side note, the first clay pipe you found was 4" clay tile that was just laid end to end with no seals around the pipe, then it was covered with pea gravel. Back in the days before perforated poly or PVC they used that method to make leach fields or to install storm drains. The little gaps between the sections allowed the water in the pipe to leach into the soil the same way that modern perf pipe does.

From the description the only part of the whole installation that is code compliant is the cleanout 3' outside the wall. If you run a drain through the wall to a buried exterior line it has to be treated as a separate sewer, which means that it MUST HAVE a vent to atmosphere. If you are under the IRC you could install an through the wall vent, subject to a list of code requirements, otherwise it requires a vent through the roof. Judging from your description even the vent on the main stack does not meet code.

Majority of washers nowdays have electric drain pumps, not driven by the main motor either via belt or directly, and drain slower than older machines. Frontloaders, of course, also involve a lesser volume of water. I recall when WP direct-drive washers were new on the market and we (the appliance service where I worked) delivered them to customers who had an older belt-drive machine ... the direct-drives discharge at a higher flow rate and the drain standpipe would sometimes overflow during the installation test-run. A workaround during the test was to set it to spin on low speed for a slower discharge rate. We'd advise the customer their drain must be serviced or reworked.