Yes...and no....
Yes, you can build quite elaborate musical compositions using a mouse to enter one note at a time into staff or PRV ... of course it does take time and a fairly decent working knowledge of musical theory and composition is really helpful.
No, you can not hit record, and then mouse in the notes and have them recorded. Nor can that be done using the on screen keyboard. To record midi in real time, you must use some sort of midi controller or a keyboard style synth plugged in to the computer or interface.
Once you have a note in the sequencer, you are either in PRV or staff view. If the note is wrong or needs editing, in PRV you can click and drag it, and grab the tail end to drag it out longer or shorter. Up and down is just as easy.
In STAFF, ... simply right click the note and a note property window opens where you can edit all the values for the note..... in staff you can also drag the note (as is) in time and pitch.. R or L... up or down.... in the scale and key you're in.
I like to use a dedicated musical keyboard..... however, I have heard a number of very well composed songs, mostly symphonic/orchestral in nature, which the composers claim were composed one note at a time, one track at a time, all manually with a mouse in Staff or PRV. I can take quite a bit of time and dedication to do it that way. But it is totally possible.