Does the Air Force (United States) help design their own planes?

In short: the US Air Force does not do any design work, primarily because they have deliberately decided not to maintain the necessary expertise. Other DoD entities do, to a greater or lesser extent. This has been true for the past couple decades, but was not true historically.

As other answers have stated though, a detailed answer depends what you consider "design," "aircraft," "does," and "help" to mean.

"Aircraft" is probably the easiest: it is fairly common to require specific systems be put on a platform. For example, you might see a requirement for a specific IFF transponder (or the prime contractor might select a specific IFF transponder), which might be provided as Government Furnished Equipment (GFE - essentially the government is supplying their own parts to be installed). That box is then part of the design of the aircraft, and the prime contractor is relying on the government to ensure it works as required. Small avionics boxes like that are more likely to have more design influence from the government side. So, if exerting design influence on that box counts as exerting design influence on the "aircraft," then the USAF does do design work. It is overwhelmingly common to have a random assortment of configuration items (especially avionics) provided as GFE, usually as a misguided attempt to save money.

"Does" is next. The USAF got out of the design business. I am not aware of any specific policy decision that drove this, but anecdotally it seems to have happened in the '90s. Before that, it was much more hit or miss. E.g. the SR-71 was built based on a half-page of requirements while Lockheed stiff-armed the USAF away from any design insight, But John Boyd famously drove the overall capability requirement for lightweight fighters and exerted tremendous design influence on the F-15.

"Design" and "helps" are a bit sticky. At a 50,000 foot view, the idealized modern US acquisition process goes something like this: combatant commanders (the heads of CENTCOM, PACOM, etc.) plan constantly for campaigns they might need to execute. As part of that, they note gaps in the capabilities available. E.g. maybe PACOM is concerned about pirates, and wants a capability to conduct small boat boarding operations from fast-mover aircraft. That capability need gets rounded with all the others and sent up to the Pentagon, where the Joint Requirements Oversight Council figures out how to address it (see: DOTMILPF), and racks-and-stacks it into a package of capabilities. If that package has a materiel component (i.e. if you need to buy stuff, not just re-train or whatever), it goes into the budget request and gets money allocated to it. Then the money and capability requirements go to one of the acquisition commands (AF Materiel Command, IIRC), which takes that crazy wishlist, turns it into a specification (usually. USAF does much less specifying than other branches), and goes and buys the thing specified. As part of the "goes and buys" process, the government usually takes control of various design elements after they are complete. The degree of control and degree of insight the government has varies wildly from program to program, but the AF tends to rely on prime contractors much more than the other services. Finally for any major weapon system and many other systems, there is an "operational evaluation," where real users really test it out to see that it meets the capability needs that JROC described. The feedback from that might well influence design.

The answer to your question largely depends on your definition of "design", as well as the specifics of the procurement program.

In general, the Department of Defense (DOD) does not have engineering authority over the equipment it procures. That is, Pentagon employees do not produce and sign engineering drawings for hardware built by contractors. However, almost all such data is reviewed by the DOD in detail before being implemented.

More broadly, though, the final configuration of a military aircraft is the result of a continuous back-and-forth between the supplier and the DOD. In my experience, this interface occasionally borders on collaboration. Sometimes a Pentagon requirement is broad and it is expected that the contractor use its expertise to come up with a creative implementation independently. Sometimes the DOD uses a so-called "directed solution", which tells the contractor almost exactly what the final product should look like in practice. And everything in between.

Your parenthetical actually illustrates a very important example. I spent years working on teams tasked with figuring out how to design military aircraft to be as friendly as possible to the mechanics and maintainers. Without the input of DOD mechanics in the field, we could not have done our job well, and the final design of the systems was certainly different than it would have been without that input.

While the Air Force (or other military unit) does not design their own jets, they do release specifications and have contests (e.g. the ATF that selected the F-22). This is basically the way it works (and not just for fighter procurement):

• A design specification is released, these include things like range, mission, weapons.
• A "paper" design is submitted from companies like Boeing and Lockheed Martin. These may be based off of existing designs or "clean sheet" designs.
• The military funds the development of one or two prototype aircraft from a number of paper designs.
• The military evaluates each design against the specification and selects a "winner".
• A larger contract is awarded

So it's not really up to companies like Boeing or Lockheed Martin to come up with their own designs and sell that to the military. They work under strict program requirements to develop something "to spec" and then demonstrate the capability. The military has a hand in the design process as this goes along, so they are not entirely "out of the loop".

The secondary market is different though. Boeing may sell (with the approval of the US Government) these aircraft to certain foreign military groups. These work more like selecting something from available stock in the dealer lot.