I want to highlight one of the replies here that distinguishes how the First Crusade was more noble-driven than royal-driven (see the label "Princes' Crusade"). Indeed, one misrepresentation here is that 11th century kingdoms worked like modern states (France, HRE), centralized institutions that raised armies and sent them out. It's not just that feudalism means kings have less power, but that the state apparatuses for raising an army are quite different and distributed to local figures (like lords). So if we're looking at capacities for conflict, "France and HRE's ability to raise armies" (as if the state is doing the action) should be clarified as the monarch's ability to organize armies and perhaps expanded to regional and local leaders' ability to raise armies and engage in conflict, since in the absence of strong state apparatuses conflict comes from leaders creating coalitions of followers and raising people loyal to them.

But the short answer to the first question is "no." For instance, Robert II (Normandy) returns from the Crusades and almost immediately tries to take the throne of England from King Henry I. They settle the dispute diplomatically after Robert lands in England (the Treaty of Alton, 1101), but the reasons for Robert settling are more likely related to Henry's popularity among the English nobles and the church than any shortage of troops. Indeed, they end up fighting anyway in 1105-6, only a few years later, culminating in the Battle of Tinchebray. Robert lost, but he wasn't necessarily short-handed. So whatever the exact numbers would've been, leaders of the time maintained their capacity to wage war almost immediately after the First Crusade.