I wouldn't touch the spindle until you have some significant experience on the lathe. Mostly because spare parts for these old machines are mostly non existent- if you mess it up you will need to machine an entirely new replacement. A big job to get exactly right. You could take the spindle out to a shop and have it repaired, or put it back together as is and see how it runs.
Regarding welding up the taper and recutting it, likely any welding that occurs will distort the overall piece, possibly twisting the axis out of alignment- resulting in worse run out at the end causing outmod round holes when drilling, and broken drill bits. So again, interview a shop or two and see what they say.
Regarding your table. It looks to be in remarkable good shape. You could leave it alone, just lightly clean off any rust and oxidation and be good to go. Or there are several methods for repairing dings, such as drilling out the hole and driving an interference fit steel rod into the newly drilled hole. Cut and mill or file flush. Reportedly this method gives good results with almost no noticeable repair being visible on the surface of the table.
regarding welding your table, I would only do so if you find a welder with actual experience welding cast iron. you need to fill up the holes with similar cast iron material or, use a nickel based rod designed specifically for cast iron. Both are very spendy. The usual modern welding rods found in most welding shops today will NOT be appropriate for the repair you want to make. You might see if you can find a cast iron repair shop nearby, if you talk to a knowledable welding supply company. Welding cast iron is mostly a lost art these days, but a few old timers still are in business. It takes a tremendous supply of evenly distributed heat to controll metal flow into a cast iron part, and a very experienced hand to get it right. To much heat will melt the surrounding cast iron and crater the hole into an increasingly bigger mess. Then when done, you also need to post heat and hold temperature on the repair, and surrounding casting, to keep it from cracking as it cools and the chemical composition of the casting changes. Hate to see you take an excellent original Buffalo table and have some guy ruin it trying to weld up a couple of minor dings.
Overall it looks like you have a very nice example of a fine old drill press! Good luck with it and have fun with the restoration.