How do my rover logs and their associated grid squares or other geographical data stay associated in LoTW?
You can submit multiple logs into LoTW, i.e., logs from each grid in which you operated, as long as you used the same callsign. In TQSL you can create as many different station locations (the QTH from where you operated) as you need. You will need to consolidate those QSOs from each different location, however. Once you have parsed your log correctly, you then sign those log extracts with your primary certificate, but make sure you select the correct station location for that particular log extract.