The Executive can't actually cut approriated federal funding, since budgets are congress's job.

The executive, in fact, must spend money that congress appropriates. Unless it is illegal/et al to do so, or the funding otherwise allows prseidential discretion, they are required to do so.

Yes, they did some EO's purporting to cut funding. None that related to non-discretionary funding have been upheld, even by "trump" judges, and so far all non-discretionary (IE explicitly directed by congress) funding cut has been restored, AFAIK. All are a wildly clear violation of separation of powers, and so far no judge has disagreed.

(Though don't confuse whether they have to spend the money the way congress directs with whether they can or can't fire federal employees, etc)

There is a path to the president impounding appropriated money through the impoundment control act, but they haven't done it or followed the process so far.