Enjoyable, but not the “I wanna have the people inside the store I’m pulling up to to hear my car fucking vibrating” level of bass.
Sub bass quality, and I couldn’t say I get what you mean by SQ
In short “SQ Subwoofer (one that is designed for sound quality) is able to reproduce a linear frequency response in a generous frequency range, while maintaining excellent transient response and transparency.”
http://xtremerevolution.net/what-makes-an-sq-subwoofer/
Just to give more information.
Best ever sound quality subs i had was from Focal and DLS. From the competition years.
Today i have a budget build with Alpine R series.
When things where more in loud department, choice was Ground Zero, SPL subs.
polks are good. the ring radiator models will give you better highs.
Hmm… Interesting. A 10" would best seem to fit what I would probably prefer. As I don’t really plan to have anyone sitting in the fold down seats, I’d probably be treating the back speakers more like rear channels, and the sub would probably go right behind the drivers seat. (Granted the Polk MM 1042 DVC is ~$20-30 more, the fact that it seems more for exterior use where it is going to be in contact with the elements… Loses more of my attention.)
The MTX 5510-44 has the appeal of being only $130 with the ability to go in a fairly small enclosure.
The Alpine R-W10D2 and R-W10D4 are only $100 more (slippery slope, I know) and the price/performance is a big thing personally. However, then there’s the “well, do I go for the 2oHm or the 4oHm?” that adds confusion.
The Focal Performance Sub P 25F is out of stock(discontinued on Crutchfield), and only has a single unit for $350 on Amazon. The Focal P 25 FE on the other hand is $350 new ($300 scratch and dent, making it more of a contender), but I have no idea if there is much(if any) difference, or how they compare.
the higher the impedance the more control the woofer has typically. the lower, the more SPL. by sounds of your situation, i would go for the dual 4 ohm.
that polk is a marine subwoofer. the cheaper polks w/o the slots are the same, but without the water proofing.
but i would look at the JL audio offerings for a subwoofer. JL audio is a highly highly acclaimed subwoofer company.
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edit: those are highly regarded for their fidelity.
If I could make the budget work for something like this, would it possibly be better?
*edit: provided it comes back into stock
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that should rock your socks off.
edit: make sure you get a beefy amp for it, dual 8ohms is a lot.
The dual 4 ohm is in stock. Possibly a slightly better option, if trying to keep overall costs a bit lower is a concern?
yes. plenty of 2 channel 4ohm amps to power it. they would power the 8ohm version as well, just at half power best case most likely. so then your looking at 500watt car amps(typically rated for 2 or 4ohm) down to 200-250watts.
What would be a recommended baseline for a 2x 8ohm voice coil sub like the 10TW3-D8 linked?
Please tell me I wouldn’t be looking at a monstrosity like this…
its 4 channel amp that can be bridged to 2 channels. stable on 8 ohm loads. should give 300+watts on each channel when bridged into 8 ohms.
Damn… Admittedly… looks complex
its not that bad. you only hook up 1 set of wires to the amp for each channel. no extra wires. think of it as a 2 channel amp for your purposes. that subwoofer has dual voice coils, so it has 2 sets of connections, one for each coil. its like hooking up stereo speakers, but just to one speaker.
Let me make sure I’m comprehending correctly:
I would only be using two channels in my use case, only using a set of speaker cables going to the terminals for each voice coil.
Now, provided that is correct, is this including bridging channels?
yes. in bridged mode, you will use the positive of 1 channel and the negative of another channel, bridging the 2 channels.
edit: typically they are listed on the terminals, but that one doesnt for some reason. the manual will let you know which ones. its typically, but not always, the far left and far right in each 2 channel set
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Anything special in regards to how it’ll need to be connected to the stereo?
depends on your stereo. but, i have not done any installations myself. i have built sub boxes for buddies. but others have always installed. sorry. pretty sure its fairly simple though. got to be a ton of youtube videos showing how to integrate subs into a car stereo. looks pretty strait forward though. most of the amps have rca and high lv.
edit: looks like most stereos have rca outputs. and i am seeing sub outs on models as well.
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Help 
What are good cheap subs and encloure brands on amazon. Have a compact JBL 40w 4ch 140w 4ohm 2x bridged amp got the 2-way speakers. Can i get away with like kicker ? Its going in my extra cap taco.
Also what you all think about compact active 8in underseat subs?
I just want to get some full range.