1. The wheel description doesn't give much detailed information but I have to presume the rear hub width is 130 mm and it accepts Shimano 8/9/10-speed cassettes. It will accept the 7-speed cassette you show but will require a 4.5 mm spacer as 8 to 10-speed cassettes and their compatible freehub bodies are wider than 7-speed.

2a. Classic from when, the '60's, 70's, '80's? Depending on it's age your frame may have 120 mm or 126 mm rear dropout spacing and may have to be either cold-set (i.e. bent) to accept a 130 mm hub or you will have to struggle to fit it each time.

2b. The switch from 27" (ISO 630) to 700c (ISO 622) wheels will require lowering the brake shoes 4 mm so the compatibility will depend more on the brakes than the frame.