I know little about all of this, so, go easy on me. I've got a mountain bike, fairly light in weight and rarely ever used for anything other than road use. The current tires say 58-559 / 26x2.35. Is it possible to go for thinner tires with the same Rim and how much will this affect the ride?
What you want for road use is slick tires -- tread and knobs are bad for road use. You have 26" (ISO 559) rims, so you need 26 x (something) tires where (something) is a number in decimal form (e.g. 1.75).
Going for smaller tires will lower the bike a bit, and smaller tires have to be run at higher pressure (so you'll get less cushioning). There will also be changes in rolling resistance (You may want to read the whole "Tech Info" column on the right side of the Schwalbe page).
As for how small you can go, technically you can mount pretty much any 26x(something in decimal) tire on, but the rim width should ideally be of the right width so the tire doesn't roll off or increase the risk of flats or rim damage.
I'd probably look at something with a puncture protection belt, like the Schwalbe Marathon Plus (26 x 2.00 will likely be one of the larger slicks they sell, and you can likely go narrower depending on the rim width).
It is possible. For typical mountain rims, the low limit is somewhere around 28mm. Some differences from mounting narrower tires are following:
Less cushioning from tires: Smaller tires can not absorb as much shock from from curbs, cracks in the pavement, etc. On the other hand, smaller tires can be made with more flexible casing and absorb small vibrations better.
Quicker steering: With wide tires friction resists steering more than with narrower ones. Depending on the geometry, the bike will be either more agile or feel nervous. Smaller rotational inertia from lighter tires will add to the feeling.
Rolling resistance: It is true that for similar tires with similar pressure the wider one has lower rolling resistance. However, narrower tires can be made more flexible and use higher air pressure so the difference is not that great. For example, compare the numbers for 4 bars with narrow tires and 2.5 bars for wide ones: https://www.fietsersbond.nl/sites/default/files/test_schwalbe.pdf
Reduced height: The entire bike will be a centimeter or two lower
Looks: Large empty spaces in forks can look strange to some people.