Customers dont place limit orders or do they?
No, they don't. In an electronic RFQ market, the requesting participant (presumably the "customer" you are referring to) is generally not obligated to show its side. This design dates back to times when you had to trade over the phone, where you would typically ask for a two-sided quote so your counterparty can't coordinate with their associates to lean against your side preference.
If you have mid price for rfq market in fixed income. Customers dont place limit orders or do they?
If you have mid price for rfq market in fixed income.
This will be platform-specific and can't be answered without knowing the specific platform. On some platforms, the responses are anonymous and matched on the same central limit order book. In such cases, there is nothing fundamentally distinguishable about a response to a RFQ from a limit order. In others, there could be a separate taker API and liquidity providers may not match with one another. If this is an ECN for OTC, there could be credit counterparty arrangements that prevent the quotes from being consolidated onto the same "book".
How does the dealer price new orders from this internal order book at a very high level?
It depends on the exact product that you are pricing, your pricing model, and how you inventory risk and less so the RFQ (or its publication) mechanism.
If there is no comparable product, you might price it against a basket of comparable products whose weights are fitted. Or there's always the main benchmark rates that get published at regular intervals from which prices can be constructed by first principles: SONIA, SOFR, TONAR etc. get published at 9 AM in their corresponding time zones. ICE has LIBOR. Refinitiv publishes theirs at 4 PM ET. OTC markets have active quotes on euro midswaps, gilts and yen midswaps.
Often in a very illiquid product (say corporate bonds, interest rate swaps), hedging out instantly is not possible so a substantiative carry needs to be priced into the spread. Moreover, in the absence of a market, one might conceivably have to hold on to a position for a significant time, so there could be a litigation release clause that allows counterparties to nullify a trade.