Hi PageMossFarm and welcome to the Forum - and to the wonderful world of keeping chickens.
There are quite a few threads in the section on runs and equipment which address your questions in detail - we love talking about runs and coops and have a lot of hard-won experience between us, so I suggest a good browse. But basically they will all conclude that the setup you suggest will be far too small for 3-4 hens, especially as you will be keeping them in there for much of the time, and very probably nearly all the time in the short, dark, wet days of winter. Chickens need a minimum of 2 sq. metres of floor run space per bird, and this is even more important in relatively small runs where there is less space in total for them to move around, and where a higher proportion of floor space will be taken up by the feeder, drinker and dustbath. There are a lot of coop+run combos on the market, mainly imported and far too small for the number of birds they claim to accommodate, both in the run and inside the coop. You’re welcome to post links to the ones you’re looking at for us to see, so people can advise. It’s really important to get this right, so your first experiences with chickens are good ones. Your best bet, for 4 hens, is to get a freestanding coop within a fixed run of at least 8 sq. metres floor space - more is good - nobody ever complains about their run being too big for the birds! Hens like social distancing more than we do.
They also poo a LOT, and in an undersized run this gets messy even if you poo pick daily. For this reason, and so that you can get in and walk around comfortably, yes you’re right, you will need a run with a high enough roof to walk into and move around. Otherwise, you won’t be able to catch your hens when you need to, or get easily to equipment to clean and refill it. Your query about vertical space applies to you, not the hens!
A roof of some sort is very desirable, especially in winter as heavy rain will make the run floor muddy and messy for both you and the birds.
I expect others will be happy to welcome you, and to help with things I’ve missed out. Looking forward to hearing how you get on - you’re so sensible to think about what’s needed before actually getting going on your project. Good luck and please keep in touch!