Relaxing on a fixed gear takes a different form than on a geared bike. You will be reminded immediately if you try and coast. Relaxing means not going as fast. You will find that you will get used to not coasting. Going fast uses energy, even if it is down hill. If I am tired, I will slow at the top of a hill and descend slower than I would normally. I may back pedal or drag the brake to keep speed in check. This will be as close as you can get to relax.
Really steep hills require planning. You never want to write a check at the top of a hill that your legs can't cash at the bottom. You really want the to be working the pedals on the decent and not having them work you. Keep your speed in check from the top. If you get going too fast, then pedal harder and hit the rear brake. This will stabilize the bike and allow you to gain control over the cadence.
In time you will find your legs will be able to tolerate a wider cadence than they did with a geared bike. I am 60 and I can spin up to 180 rpm routinely on hills and 200+ in a pinch. When I rode a geared bike I probably never got over 150.
Up hills are another issue. You get better at just grunting it out on hills. You do need to select gears that make it possible to ride on the steepest hills in your area. I get by with 70" gearing and manage short 10% hills routinely. I would start with gearing that is a bit shorter and get a feel for things to start.
Ride Safe,
Joe