First, expose the pipe and find out what material is used. Only, then you can you decide how to patch or replace it. It may be PVC, ABS, bituminous fiber, clay or cast iron. At that time, you might want to get in a plumber to do the actual work. The biggest cost is the digging. For safety, work with a large sloped excavation and brace it. Here is a link that might be useful: Excavation safety

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.

Randy427 addresses the first concerns, whether you have proper drainage and/or hydrostatic problems. Block walls tend to leak. Digging up the outside and parging,along with subsoil drainage, would eliminate infiltration. While you're at it, some rigid foam could be installed. If your intent is to simply saw off the pipe, I'd suggest that instead you caulk/seal/paint around the pipe to close off the foam.

Sympathy extended. If you have a septic tank and also have flush issues, consider having the tank and lines pumped out/emptied. Even a small leak over time can "liquify" surrounding dirt ( create muddy water) and move it, including into the pipe, where it can settle and again become mud. Consider replacing the pipe where any specific clog has been found. That said, the junk -- nails, etc., -- being left in the pump were not a "natural" event and may even have damaged the pump. Your contractor and his plumber(s) need to make this right asap. Some tree roots (especially water friendly ones such as weeping willow) can enter and fill up and stop up septic drain fields. Ditto that for any other drain fields such as sump pump removing rainwater from a basement. Would not have any sump pump that is intended to remove only rain water ( no sewage) be emptied into any septic system. Rainwater runoff needs to be discharged into the yard/roadside ditch. Would not want a sump pump as a default -- would want the exterior and interior of any basement wall waterproofed so no sump pump is needed. That moisture is still in the concrete that has not yet seeped into the basement and puddled on the floor IS still moisture your home will be absorbing, potentially causing health threatening mold and mildew. Do you have a lawyer you've used before/trust -- such as for your real estate closing? Not a lawyer but you may well need to talk with one about protecting yourselves in this situation. Don't threaten getting a lawyer or going to court unless until your lawyer says it's actually in your best interest and the right time to do that. Instead, gather all documents you've signed regarding this build/purchase and document the problems (photographs -- dated notes). In my views, given the sump pump failure, you should consider asking (courteously demanding) waterproofing outside your basement/cellar/foundation wall so the sump pump shouldn't be needed and, if necessary, even seek to withhold payment until the issue is addressed -- but ask your lawyer. Check about waterproofing the exterior of your basement/cellar walls -- requiring digging down beside them -- or at least waterproofing inside the wall.. Check about better landscaping with the objective of "turning the rainwater" away from your basement/cellar wall. Good luck.