Theoretically, if all firms are identical and have the same cost schedule, then the supply curve for the market should be perfectly elastic, as you stated. The side-by-side market and firm graphs would appear as follows:
This is an odd situation. Typically, when there exists a surplus of a good, firms competitively lower their prices until enough firms go out of business for the surplus to no longer exist; and when there exists a shortage, firms enter the industry until the supply increases enough for the shortage to vanish.
However, in this case, since firms are not distinguished by the price that they can afford to charge, the natural tendency towards equilibrium cannot occur this way. Instead, when there is a surplus, firms cannot compete by price lowering and will either survive or perish based on the random purchase choices of buyers (if prices are the same everywhere, consumers will probably choose where they buy their goods based on other factors, like geographic proximity, aesthetic preferences, etc, but “rational” buyers in the simplified model will be indifferent). For our purposes, which firms survive and which close down will be essentially random - whichever firms happen not to sell enough of their product for whatever reason will close.
During a shortage, it gets even weirder. Whether or not firms enter the industry depends on the location of the ATC curve (which is why I did not draw it in the above graphs). If $ATC < P_F$ at $Q_F$, firms would enter since profits are possible and people will buy the good they produce. If $ATC = P_F$ at $Q_F$, there is no telling whether firms will enter, since they will all be indifferent.
And if $ATC > P_F$ at $Q_F$, whether there is a surplus or shortage, firms will drop dead like flies until not a one is left. If the good being produced is of any real import, the government would probably offer subsidies to incentivise the production of the good.
So, to answer your question - it would make sense for you (a firm) to enter if $Q<Q_M$ and $ATC < P_F$ at $Q_F$. You could also enter when $Q\ge Q_M$, but then your success or failure would be left up to chance or other factors as consumers decide whether to buy your product or your competitor’s identically-priced product.