How do weekly seminar speakers get picked by a STEM department? When there's a weekly seminar in a STEM department that professors and grad students attend, how are these speakers given slots?
Are they invited?  By whom?
Can they apply to speak at some weekly seminar, e.g. apply to give a math talk at Harvard?
Do the seminar speakers then get their airfare and hotel paid for by the department's own budget?
As has been noted, everywhere will have its own procedures, but I think my department is fairly typical of what I've seen elsewhere:

One or two junior faculty are responsible for organising the main seminar series
Every few months they solicit suggestions for seminar speakers from the other faculty. Nominations come in different forms, e.g.:

&quot;I think Prof. Jones work is cool, we should invite her sometime&quot;
&quot;Prof. Jones is going to be visiting me for two weeks in March, perhaps she could give a talk&quot;
&quot;Prof. Jones emailed me to say she's going to be in our state for that conference in May, maybe we could get her up here too?&quot;.


The seminar organisers then try and assemble a programme, balancing many factors (breadth of topics, speaker availability, diversity considerations, political clout of nominator...)
The department expects to pay some travel and a night or two's accommodation for anyone scheduled in the main seminar series. Sometimes speakers will decline this (e.g. if they were in town anyway); sometimes individual faculty will supplement it (e.g. to enable someone to stay for a whole week).

In addition to this main seminar series, individual faculty members may arrange 'special seminars'. Typically this occurs when someone happens to be visiting for other reasons, and there isn't a convenient empty slot in the main seminar series. Special seminars may not be as widely-advertised or attended as the main series. Any costs are typically borne by the hosting faculty member (out of their grants/etc), rather than the department (but they are often minimal, as the speaker was invariably in town anyway).
I think it is pretty uncommon to specifically offer to contribute to the department's main seminar series. However, it is very common to contact acquaintances with messages like &quot;Hi Prof Smith, I'm going to be in town next month. If you have time, it would be nice to catch up and I can show you my new proof of the Basketweaver Theorem. Of course, I'd be happy to make it into a talk!&quot;. In such circumstances, you should not then expect any reimbursements, though often you will at least get taken out for lunch/dinner.