I ran baseboard for years as Casey described... measure your cuts, make a list, go to remote cutting area, go back and nail. We also used to use this same method on a two man production team, one guy "nailer" makes list for room or next few sections, hands list to "cutter", cutter brings back and scatters ahead of nailer.. repeat/repeat/repeat to do a complete house. Same method works equally well for casing and crown molding. On new construction as well as major remodels we would typically have a "shop" set up in a garage or large room for all cutting and sawdust, this way painters as well as other trades can continue working...

You could try searching for "stair barrier for small dogs" or "pet safety net for stairs" to see if you can find any products that might work for you. One option could be to use clear plastic carpet runners or rug grippers on the stairs to cover the gaps between the treads. You could also consider installing baby gates at the top and bottom of the stairs to further prevent your dogs from accessing them. Another idea would be to hire a contractor or handyman to install custom-made nets or barriers to fit your specific staircase. They may be able to provide you with more personalized options to ensure your dogs' safety.

If you insist on the lighter flooring, then I would go with the slightly darker version (the one on the left in the last photo). Are you keeping the brick? If you are, the slightly darker wood floors are a better compliment than the super light wood. If you are removing the brick and rebuilding the stairs, then you will want to find out if either of the vinyl floors have stair treads and nosing available. People who have used vinyl that does NOT have treads/nosing found their BIGGEST headache comes from trying to find a solution to this problem. As for your darker wood floors, they look (just one photo with one chip out of the wood) like they could be ready for a sand/refinish. If they are, I HIGHLY recommend doing the hardwood FIRST...and then choosing the fake-wood look product for the den. You might find a good colour match can be had if your red oak hardwoods can be sanded and refinished as 'raw' = the colour of the planks you want to use. How old are your hardwood floors? If the finish is very close to 20 years old, they are ready for a full sand/refinish. It is always advisable to figure out the BIG ticket item before you try to figure out a small ticket item. And if you are refinishing the hardwood, why not add more of the same??

Shoe molding. Must be vintage stairs: way out of code. Too steep.