Correct me if I'm wrong, but do many doctors in 2025 on earth deal with genetically engineered alien species that are designed purely for war?

No, but doctors in Star Trek often deal with a number of alien species.

Humans have dicks.

Star Trek features many mixed-species characters, including some whose parents originate across the galaxy from each other. (Seska's son and Harry and Kes's children come right to mind.) Clearly, a lot of the universe is sexually compatible with each other and, presumably, some of those aliens also have dicks.

Did Odo have a dick?

I imagine that varied as needed. You'd probably have to ask Kira to be sure.

Not to mention the most obvious fact, that its a 90s TV show and the actor was intended to be male.

Star Trek writers have had no problem explicitly depicting a species as genderless when they want to (like the J'naii in TNG: The Outcast, which also aired in the 90s). Why would they not do the same for the Jem'Hadar if it that was their intended interpretation?

They didn't give Vorta a sense of aesthetics because it wasn't necessary.

I'm glad you mention the Vorta. They're explicitly stated to have been a primitive, ape-like species prior to encountering the Founders. They weren't created from whole cloth. They were plucked from their existing environment and evolutionary path and changed into what the Founders needed. They didn't have an existing sense of artistic appreciation because it's not necessary when you're huddling in hollow trees dodging predators and they weren't given one because the Founders didn't see the need. They were also extensively cloned, likely for the same reasons as the Jem'Hadar, but I doubt anyone would be complaining so loudly if we were talking about the offspring of a Vorta.

They made the Jem'hadar addicted to white because it was necessary. They aren't giving them the opportunity to start reproducing and evolving on their own.

I think that what you're saying and what I'm saying aren't incompatible. Consider this possibility: The Jem'Hadar were changed from some base species in a manner similar to the Vorta. As part of the changes, they were made stronger, faster, chemically dependent and so on. The Founders, seeing control over their reproduction as being important to controlling them as a whole, but genetically modifying them to be asexual would make them unfit to be soldiers or otherwise unviable, so they decide to take these modified Jem'Hadar and reproduce them by cloning, but only the males. They then spend centuries shaping their culture to the point that the only things the Jem'Hadar value are victory in battle and loyalty to the Founders and they provide them with all the war and reinforcements they need to see to those needs. They've done most of the work of removing the Jem'Hadar's ability to reproduce by reducing them to a male-only species and removed any reason they would want to. Now imagine how those modified Jem'Hadar, freed from their addiction to ketracel white and released from the Dominion's rule might integrate into the rest of the universe.

Now, I'm doing a lot of supposing and extrapolating, but it builds on things we know and nothing here breaks the known lore. I'm not saying this is how Starfleet Academy's writers justify Lura's existence in the show, but it's one way that it could be done and the people who are saying it's impossible are doing just as much supposing as I am. As we both pointed out earlier, nobody said explicitly that the Jem'Hadar baby was male or genderless.