Since a broker does not allow its customers to quote two sides, can one quote bids via one broker and asks via another broker then? Is it "illegal"?
You can quote both sides, just not at the same price. In general, avoid manipulation, avoid wash sales, avoid bad-faith bids and offers, and you'll be fine.
But what if you quote the offer side, the bots follow you, and then you hit their offer on a different exchange. Technically you are never showing a quote because your order is marketable. This whole "manipulation" nonsense should just be thrown out the window. Let people trade. If you are a bot and stupid enough to follow on the offer, you deserve what you get.
You are allowed on one exchange to BTO and STC, but not BTO and STO. THere is no restriction to do that on two exchanges that I know of except it is not practical as each order must be tagged correctly. E.G. buy to open 10 calls at 1.00, sell to open 10 calls at 1.05. Once you buy 1 contract, the sell 10 lot is mismarked at it now should be sell 1 to close, sell 9 to open. The entire offer needs to be canceled. This in one reason many brokers do not allow this as it is hard to code and be compliant. Market makers are not required to be marked open/close, so they are exempt.
That is not allowed as far as I know. I also, trade options. Sold a call option one time when my intention was to buy the call option. To close it, I tried buying the call option at double the contracts and my order was rejected. So, I just closed the initial position. Then, placed a 2nd order to buy the call option and was filled in seconds. As to market makers shadowing you, that they do! The remedy is to outsmart them. When you find the bid and asked spreads are very wide and getting wider as you try and fill your order, I would just assume that there is no "serious buyer or seller" and that all you have are market makers low balling you to get your options at the cheapest price. You realise the next day, they will do the opposite and bid the price higher to sell you that same option. So, if there are no serious buyers or sellers at fair prices, cancel your order and wait they next day. Chances are good, the option prices will be higher as market makers drive the prices higher!