I spent (wasted?) most of my life in machine shops, early on I worked for Sheldon Lathe until they went bankrupt in the early 80's. I only mention this as a qualifier for my opinions, there are very few people I'd call "experts" at machine tool building so take all comments/replies as opinions that may be correct for their circumstances but not necessarily correct for what you want/need to do.
The dings to the bed-way may/may not affect accuracy but why leave them in there? Other comments about wear will indeed be a bigger factor in accuracy but there's little you can do about it right now besides learning compensating techniques we needn't go into just now.
The comments about taking a new, clean stone to the dings is good but I'd suggest starting with a smaller (maybe 4" x 1/2") fine India oilstone (water stones have particular methods) to work the raised surfaces of "dings". When there's crater below the surface there's also displaced metal above the surface that will show up as shiny, new metal type spots when stoning. Using the larger/wider stones is good for larger surfaces like once you've removed all the minor afflictions. Using a larger stone will have a tendency to abrade areas on the bed-way that don't need to be touched and/or might be original factory surfaces. The approach you want to take is to only remove the high spots where needed instead of giving the entire bed a rubdown. You'll need to examine just where the contact surfaces for the saddle-to-bed-way are, there are other surfaces (usually fairly rough surfaces) that aren't even contacted by the saddle. No point in working on those even if they have dings in them, working on non-contact surfaces is a complete waste of time/effort. Resist the urge to use a file on ANY precision surface.
Use light to medium pressure on the oilstone with just your fingers and only where there's a raised surface (very small areas). Keep the stone wet, avoid long stoking motions that will remove some of the factory made surface. If the stone starts getting dark, shiny spots then it's getting clogged up and will require a scrubbing with a brush to clean out the pores of the stone or it won't abrade/cut like it's supposed to.
When you think you've flattened all the raised areas then you can very lightly run a larger stone (Fine India, Washita, or Arkansas if properly flattened) the length of the bed a time or two, feeling for "bumps" or a slight drag on the stone that would indicate more selective stoning is in order. I often use a black magic marker on the afflicted area as a visual aid to when I'm abrading a raised surface or when it's been flattened to be flush to the original surface. When that's all done to your satisfaction a good cleaning is called for to remove all the abrasive slurry and a light oiling of the clean surfaces. If you feel adventurous it would be a good idea to remove the saddle and the cross slide for stoning/cleaning as well. Be aware of brass gibs/set screws on the slide surfaces. Remove the tailstock for similar treatment.
The broken shaft section is another matter. I'm not familiar enough with that model to determine whether your part needs repair or replacement. Better photos enhanced with arrows/circles (photoshop) will provide better information. If you don't have the broken off ear it appears that a welded sub-assembly (two plates and a threaded bushing) might work just fine. The other model yoke my/may not fit without further dimensional information. I should also state that Logan is still in business (Logan Actuator in Harvard Illinois), sells some parts, likely has all the records/manuals for your lathe. Repair parts where you can but some items (usually the bull gear) should be replaced if a qualified repair cannot be attempted. Logan made a decent quality hobby grade lathe for homeowners/farm use. It appears as though your model has thread cutting capacity, that's a real advantage over some industrial grade lathes that cannot (Hardinge for example). Your Logan should provide many years of capable service, especially for entry into metal working, depending upon the tolerance-per-length you hope to achieve. Hope you got some tooling with it that will open possibilities for you.
Do please pardon the length of my reply and the following statements that must be repeated. The lathe can be one of the more dangerous machines in the shop. There are safety HABITS you need to learn and understand to avoid bodily injury. The tiny 7 x 12 lathes from Horrible Freight will break fingers and/or gash into flesh. Your lathe is much larger, more powerful, and is capable pulling you into rotation that could result in being maimed/blinded/disfigured for life. Reading material on lathe operation is recommend and SHOULD have at least one section on safety. Stories of a guy getting sucked into the spindle are real, I have either witnessed or been a part of those stories. Safety glasses ALWAYS on, no long sleeves or gloves, no watches/rings, hair tightly secured in place, wait for the spindle to come to a stop before doing ANYTHING near it, stand clear of the debris field throw, chew your food 20 times, wear your galoshes if it's raining.........you get the idea. Do take safety seriously, it's not a matter of IF you'll get injured it's WHEN and how bad the injury. It happens to everybody but you can lessen the result by adopting safe working HABITS that will limit how serious the injury is. Take for example, pulling the tangled "birds-nest" of metal peeling off your workpiece. Resist the temptation to reach in to pull it out. Whether with pliers, a hook-on-a-handle, or a brush, it can/will get grabbed/pulled into the spinning chuck or the long stringers can curl/loop around your fingers/hand. All machines must come to a full stop.
Ask a lot of questions, I had to with grumpy old men that always told to me pass their knowledge/skills along to the next generation. Now that I am Grumpy Olde Man I feel obliged to honor those that taught me. Contact me off list if nobody will talk, some don't. Welcome to the world of precision metal working, it's a large room.
