Sir, you did well. It is nothing to write home about, but should give you years of reasonable quality music reproduction at an acceptable cost. The rest of $ is better spent on the software - vinyl itself.
AT-403EP cartridge is a T4P P-Mount version of the AT 120 family - and can take all the styli in this vast family. Of current production ATN-440MLa and ATN-150MLX
are the top replacements - but JICO also makes Shibata profile styli that fit for less money.
Equalizer, if of enough high quality and used correctly, is perhaps one of the best choices you can make in an audio system. But it is true that satisfying that "if" may prove to be a bit much - do not give up if the first attempt will not sound as you intended or wished for.
Please read about test records, stylus cleaners, record cleaning machines "quite a couple posts back" in this thread. And then settle for something you are comfortable with as you progress on your analog journey .
I was actually considering an equalizer unit. I see a lot of extremely cheap 10 band ones on ebay for like 15-20$. Could one actually go wrong with these in fully functional condition? Of course I would wait a week or two for that due to logistics but the receiver unit I purchased actually has treble, bass and balance at least.