Asking to visit a university, and be given workspace for a week

I disagree with the existing answers. I don't see this as a "big ask" at all, as long as your expectations are reasonable. You should not expect to get a nice individual office, but if I know you at all I would definitely be able to find you a space somewhere in my or my division's labs. My students have also done similar visits in the past, sometimes in labs that we did not have extensive connections to, so I guess other people also don't see this as a huge deal.

Of course you should have some idea what you and the host will get out of the visit - people may not be thrilled to host you if this is really just a cover-up for a private visit. Conversely, for a week-long visit, you should also not expect to be "entertained" for the entire week. People will not be able to cancel everything for an entire week.

It's unlikely that a department would extend such a courtesy to a visiting graduate student—normally, priority is given to faculty (or equivalent). It might be possible that the research group that would be hosting you might have a free desk in their shared lab or office space, but I would think anything more than that is potentially a "big ask," as it depends on space constraints. It doesn't hurt to ask, but I would definitely ask "Do you have a workspace for visitors?" rather than "invite yourself" to an office space.

As for the issue of a weeklong stay, it's quite unusual for a grad student to stay that long, in my experience. It does happen, but usually there's an existing collaboration or connection between the host group and the guest's group. You'd probably need a good reason to justify a stay that long otherwise that goes beyond just giving a seminar.