I'm finding it a bit difficult to understand your queries. I regularly trade 1 or 2 option contracts at a time, and have for many years. I am/was unaware of any moral or legal requirement to trade a larger number of contracts for each trade. Regarding your specific example of GS, at a share price of $209.37, a one contract trade represents a stock position on exercise/assignment of over $20,000 - which isn't a small amount of money.
I often get fills comprising batches of little fills with Robinhood, but not with other brokers. I trade spreads.
How do you find your calendars? TDAmeritrade has a link that suggests several calendar spreads but I have never traded them. Do you follow the suggestions of others or do you find your own?
I have been very actively trading calendars for 2 years - I find my own, using the techniques that I have learnt in this period. Without going into details, I use a variation of the techniques practiced at Steady Options (subscription service). I have been successful with these cals on a regular and consistent fashion. However, the drawbacks are (1) high comms (2) not possible to scale up to a 6-figure portfolio (3) very time consuming - not so much of an issue for me as I trade full-time, but if I had a day-job, then I wouldn't be able to do it.
My non professional guess is that those (we small mom and pop retails trade against) nowadays are algo driven?
There are a number of reasons your 10 lot order may be filled in smaller increments. The most likely would be the pro-rata fill rules at an exchange. For example when your order is routed to an exchange, the DPM may be guaranteed a certain percent of the volume. The rest is split up based on the size quoted by other market makers. So in your example of the 10 lot, the DPM may have gotten the biggest cut, while the rest of the order was split among other MM's.