I just completed a similar project for a mobile home that is being used as an office. An employee actually fell through the floor (fortunately only 30" to the ground). The plywood had moisture damage due to a rubber back carpet that trapped moisture for many years. We pulled up the plywood in...
Remove the existing insulation and have the underside of the floor sprayed with polyurethane insulation. It provides excellent r-values and blocks out all air intrusion. I wouldn't recommend a water-blown product as these are more prone to moisture retention.
Mike
The shims under the threshold are required when the floor is not level. As others have stated, as long as the threshold is solid you will be fine. Some caulk to prevent insects from entering, a piece of trim to cover the gap and a little paint and it will look great.
Mike
For basement applications a fully adhered vinyl plank would work out nicely. Highly durable, resistant to moisture, and they have a wide selection of styles to meet your decor objectives. I have specified these floors in many commercial applications with great success. They are relatively easy...
+1 for the vinyl planking. Especially being for a rental, vinyl planking is very durable. There is a wide variety of different styles to match any taste or decor. Be sure to buy a quality product, like Armstrong. We've installed it in many high-traffic commercial applications and it's held up...
Not sure if this is too late for you but any quality latex-based sealant will be sufficient for this application. The more important component would be the type of paint that is applied over the top, as this would be the primary protection from moisture and UV. DAP ALEX Plus Acrylic Latex caulk...