8 hour layover in Frankfurt

That should be plenty of time to explore Frankfurt a bit. I did this very thing on my very first trip to Europe so many years ago.

You pass through a line to show your passport and bypass customs as you have nothing to declare--it'll take mere minutes. I would then freshen up, withdraw some cash, and get on the train.
I would give yourself about two hours for the return process.

https://www.frankfurt-airport.com/en/transport-and-parking/by-train/local-public-transportation.html

https://allaboutrosalilla.com/one-day-in-frankfurt/

https://www.dangerous-business.com/one-day-in-frankfurt/

Well, you do have time to leave the airport, although there is not all that
much to see in downtown Frankfurt IMO (others may disagree).

You could consider going to Mainz, which is a ~30 min train ride.

Or get a day room at one of the hotels (Sheraton, Hilton, Hilton Garden)
in the airport complex and relax/have a nice dinner.

As to your other questions, you wont have to do a passport check if you dont leave
the secure area of the airport. If you do, you will have to do a passport check leaving
the airport (its as if Germany is your final destination), go thru security upon your
return, and maybe go thru immigration (since your final dest is non-Schengen).

How long to clear security? Who knows? Prob not too long, but just about everyone
who travels has a horror story about Frankfurt.

I had a similar layover in Frankfurt last September. I personally would not leave the airport. First, your plane may arrive at a remote "gate" which is really just the tarmac and you will have to get on a bus. Even if you are one of the first people off the plane you have to wait until the bus is full before it leaves. And if you are on one of the later buses..... After the bus leaves, you'll get a nice behind the scenes look of the airport and arrive at the terminal which closely resembles a bus station. Assuming you are travelling on a non-EU passport the immigration lines can be very long. I think it took us 45 minutes. So when everything is said and done your layover may be closer to 6 hours. Since you're leaving the Schengen zone for Scotland you'll have another passport check to catch the onward flight and have to go through security. So now your layover is 4 hours. If it takes 30 minutes to get some place and 30 minutes to return you're left with 3 hours to explore. After a trans-Atlantic flight I wouldn't be awake enough to appreciate anything. I know it's hard to waste 6-8 hours sitting at an airport terminal but maybe you could pay to use one of the lounges and take a nap. The Lufthansa lounge is pretty nice. And if you are flying to Scotland on Lufthansa make sure you download the Lufthansa app. The gate changes for the onward flight will be numerous.

Lots of people leave the airport during a layover and yours is quite long. There is no reason to hang out at the airport when there are so many things to see in Frankfurt. Train every 15 min. and it only takes 11-15 min. to get into the city center of Frankfurt.
Security is quick, and you can leave your carry on at the luggage storage facility. 10.50€ for the day.

What are your interests? The day makes a difference. Great farmers markets in downtown Frankfurt on Thur, Fri. and Sat. and the Kleinmarkthalle is open 6 days a week starting at 08:00. Medieval churches, Jewish cultural sites, interesting architecture, the new Alt Stadt. You could also take the bus from the airport to Höchst and visit one of the oldest churches in Germany, see 100s of half-timbered houses, a small schloss with a dry moat and a wonderful Farmers market on Tues. Fri. and Sat. from 07:00-13:00

Go to Mainz instead of Frankfurt.

Check out TripAdvisor under things to do in Mainz and Frankfurt.