If there are larger gaps at hatches I've seen carpet rolled over the edges and glued or stapled.
If close fit and you just cut edge clean you can stop any travelling by using a hit melt glue gun on the edge of the backing. I've applied hot melt and smooth / work it into the backing w a putty knife.
I'd say tuck under & staple could work if installing padding and you cut padding back in that edge are... otherwise I think the extra thickness will be evident?
I hope she made the "change" before you took up the padding and removed the glue.
Ideally, you still have the removed carpet around so you can use it as a pattern. If not, butcher paper may work to make the pattern.
I had a company replace my carpet. I watched. Carpet was 'there' so it will be easy to figure out how to do the hatches. Chances are, there is no padding around the edges of the hatch covers. Glue and tuck and staple the carpet around and under the edges. Yes, it will stick up but, a week or so of normal walking on the carpeted hatch covers will compress the carpet and make a flat surface.
When my carpet was remove, there was some thin 'rubber' backing (padding) glued between the actual floor and the back of the padding and then, on top of the padding. I walked on the sticky surface for a week while the carpet folks cut and prepared the carpet. Nothing detrimental to glue surface but, I cant say that about the socks I wore while waiting for the carpet installation.
Get some pieces cut and bound for heavily trafficed areas. Careful with these, we dont want you to trip and fall. Best to remove and store these pieces while underway.
For the galley area... after carpeting, I visited William Sanoma, purchased a long spongy rubber runner and cut to fit. Yup, expensive at the store.... Took a while to get used to the spongy effect but, spending time in the galley preparing a big meal will make you appreciate the effect on your legs and far more easier to clean after you drop food bits and pieces and grease splatters on it.
I had considered not carpeting the steps but decided against it because the carpet is there for sound deadening too. I still have that option. Maybe just leaving for the next owner to decide.