Ok cool. So with that budget you have the option to build something pretty high end, but I'm going to make some suggestions that might be less popular so I'm hoping other people chime in to offer alternative viewpoints.
First, the wheelbase. With your budget you could probably afford to go straight to Direct Drive here with something like either a Simucube 2 or Fanatec Podium DD. Just keep in mind that wheels for the SC2 (and aftermarket wheels for the Podium) tend to be much more expensive so you need to budget for that also. However, I'm going to make the first controversial statement and suggest maybe looking at something like a Fanatec CSW 2.5 instead. It is 1/3 to 1/2 of the cost of the DD wheels and is still an extremely high performance base; it was the top-end base before the DD days. Fanatec has plenty of decent wheels available for it for fairly reasonable prices also. I have an E30 race car that I think would have fairly similar steering effort to your Spec Miata, and I'm currently using a Thrustmaster TS-PC wheel, which is a step below the CSW, and even that easily provides as much steering force as my race car. I like my TS-PC, but if I were doing it again I would probably just go straight to the CSW, it's not much more money for a noticeable step up in my opinion. The DD wheels do provide some more fidelity, and a huge amount more force, but they also feel a little funky around center compared to a belt driven wheel in my opinion.
Second, the pedals. This is a tough one. I think a good brake pedal is almost as important as the wheel base for outright speed, but you also mention you want it to feel as much like your Miata as possible. The thing is, a lot of the really high-end pedal sets have extremely firm brake pedals that don't feel much like a production-based sports car pedal. So, with that in mind, I'm going to make another controversial statement and say at least consider the Fanatec CSL Elite LC pedal set. This is probably the cheapest loadcell pedal set on the market, but it is really pretty solid. The reason I mention it is because with the medium bushings (or even more with the soft, but I find those to be too much of a compromise) and the little filler bushing in, you actually get a decent amount of pedal travel and a squishier pedal feel that I think more closely mimics a production based car. However I may be nuts and you should just go straight to the Clubsport V3s or Heusinkveld Sprints and be done with it. No matter what, you could look at mounting all these pedals upside down to more closely mimic your Miata since they are all bottom-hinged pedal sets and I believe all 3 pedals on Miatas are top-hinged if I remember correctly.
Shifter: There aren't a huge number of options, but I think a Fanatec Clubsport SQ 1.5 shifter is probably the way to go. It's fine, and anything better than this and you just get into paying obscene amounts of money for something that doesn't have as big an impact as other things for me personally.
Rig: I think this is actually super important. My enjoyment of sim racing went way up when I built a dedicated rig. So if you have the space, I recommend an 80/20 rig of some sort. I'm personally a fan of the Sim Lab rigs, either the P1 or GT1 Evo. These will be very solid and allow you to adapt them as you add different things later. Get you a similar racing seat to what you have in your Miata and you'll be good to go.
Lastly, you don't mention this, but I do personally really enjoy having bass shakers on my rig. These could be a very low budget alternative to tide you over until you get the motion you mentioned. Some people have extremely elaborate setups, but I just have one under my pedals and one under my seat to separate the front wheel inputs from the rears, and it really adds an extra layer of immersion. We have whole giant threads on this topic with lots of debate, but I have a Buttkicker Advance under my seat and AuraSound AST-2B-4 under my pedals, driven by a Behringer NX1000D amplifier (DSP is a must no matter what amplifier you decide on) to give you some idea of what to look at for pricing. I find it to be a very good combo.
Anyway, this is a huge amount of info, so let me know if anything is unclear or you have any questions. As I said, I hope others jump in with their opinions to temper my possibly strange recommendations
