[edit2] Wow, thanks for the gold, stranger! First time I got a comment gilded :-)

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Ok, lots of questions, I'll try to hit them all. (edit: didn't realize how long this was going to be, sorry!! TL;DR: accessories are fine, but don't invest in that instrument itself because it has no resale value and if/when you upgrade, it will have good "parts" on it already).

Strings - your new strings (linked) look better than the stock ones. Not good. But a bit less bad. If you were thinking of upgrading, I would say don't get a full set. Just replace the A/D (or even just the A) and see how it compares. They should be about $20 each for the A/D strings. Those are the spares I carry in my case (I play Larsen, but don't want $50 spares, especially if someone else asks for one!). Jargar are kinda harsh, but in a good way - they are loud, easy to project, and cut through the mix of the orchestra very well. Easier to be heard, that kind of harsh. On the downside, they require more work of you to not squeak. The cheaper the strings, the easier they squeak, and crunch, and eventually go dead if you apply enough pressure (like crunching that never turns into tone?) but maybe that's not happening to you if you don't have good rosin. So onto point 2!

Rosin. Get this stuff it's superb. I used to pay 2-3x that price, I'm so thrilled it's become readily available. I'm not sure it's quite as good as it was in the 90s, but after experimenting with about a dozen brands (including synthetic, because I'm allergic to pine), this is the one I come back to. You'd think it doesn't much matter, and I used to think that too. Until I did the experiment and found that rosin brand/composition does make a difference. I don't know how much you can hear it, but you can DEFINITELY feel it! Which makes playing easier/harder, and the audience can always hear you working hard. They want to hear you make it look easy, right? Right. Get the Hill Light (violin/viola/cello). Smooth as butter, easy as pie, and creamy like a... something french dessert, I don't know... As for the old rosin, use a dry cloth and just gently wipe and wipe and wipe. Don't use alcohol. it will dry out the hair. Also, you can put new rosin even if the hair isn't 100% clean. It mixes right in, no problems. Wipe off some old, then put on too much of the new Hill rosin. It will make some dust, then in a week put on too much again. Then it should be good to apply once or twice a week from there (depending on how much you play).

Tailpiece is 100% you're done. End game quality right there. Some prefer wood, but whatever. I think I have one of those on one of my cellos. Only problem is you can't fit gut strings because they're thicker and have knots instead of balls at the end :-) The reason it's easier to tune is not the tailpiece, though, it's the fine tuners. Either way, you're sorted. Mazel tov.

Bow - holy cow, that looks legit for the price! Depends on the quality of the hair, though... I pay about that for just the hair every couple years on my carbon fiber bow (CodaBow Classic, I think they're not as expensive as they used to be, mine was top of the line 15-20 years ago and I paid like $1500 for it? Carbon Fiber was not as common so much more expensive in every product... now you can get CF sunglasses for $20 haha). Hard to tell if your struggles might be from the bow hair (stick is probably fine) or the rosin. Easier/cheaper to change the rosin first, though!

For the buildup on the strings, I'd say replace the strings (haha!) - no, but really, take a clean handkerchief and give them a rough hand-wiping at least once a week (some do it every time they play, but not me, I'm lazy!). Every now and then (once a year for me?) I'll put a little bit of rubbing alcohol on the cloth and carefully wipe the rosin-y parts of the strings. If you get alcohol on the cello, it will dissolve the varnish. That's bad. If you get it on the fingerboard, you might find out your fingerboard is painted black and not real ebony... etc. Just be careful. Also, when you clean the strings, take a few seconds to clean the entire length from nut to bridge, because oils from your skin build up in between the wrappings on the strings and make them heavier (so they don't want to vibrate as freely). This extends the life of my strings from 1 year to like 5 years. (again, I'm lazy. Sometimes I don't change for 10 years... but I like a dull/warm sound anyway).

LASTLY

On to the cello itself... Where to begin... The tuning pegs can be replaced, and the peg box can be reamed to better fit the friction pegs (or geared pegs if you want to go crazy). People used friction pegs for hundreds of years so don't moan too loudly. As for the tone, I think the thickness of the finish/varnish make a big difference, the amount of glue (and type) in the seams makes a difference, and the thickness of the wood. The thinner it is, the easier it vibrates. the heavier it is, the harder it is to get a full sweet sound. That also has some to do with the quality of the bridge (which needs to be carved by a luthier, you can't just get a Fournier bridge from Amazon and stick it on there... Expect to pay $75-250 for a bridge out the door from a pro, depending on the quality of the blank. He will also set it up for you). Don't put a fancy bridge on a junk cello tho. It will always be heavy. And it feels heavy when you play. Even in just the left hand. The neck is chunky like a baseball bat (so it won't break?) and the fingerboard might not be perfectly planed, the action can be too high (bridge height can help this, but the nut can also be a problem - and at that point, you do get buzz sometimes if the fingerboard isn't perfectly flat). The fingerboard can be thick/heavy and not vibrate easily - which means it absorbs vibration from the strings, and deadens your sound. Mostly, it's the playability and feel - not like sharp frets, more like the smoothness of the neck like a Strat with nitro finish or matte vs full gloss poly, right? People sand down the neck to get a "nicer" smoother playing? Same thing. It will also feel different in your knees (like a nicer guitar can feel better in your lap? Maybe a $3k American Strat doesn't feel any better than a $500 mexican, but it might feel a LOT better than a $50 knock-off, you know?)

I wouldn't put much more into the cello itself. I'd get a decent A string, depending on what your budget and goals are, I'd recommend the $20 Jargar or maybe D'Addario Kaplan which is much gentler sounding and feeling, sweet and easy to play, but not loud, hard to project and easy to choke it if you press too hard, but I like them for beginners overall. I'd say get good rosin. The wood bow it came with is also probably fine, but play both and see if you can actually feel a difference - play for 15 minutes with one, then switch for 15 minutes so you have time to get used to it. Put good (Hill) rosin on both of them.

Lastly, after you've been playing for a bit (enough to play a Bach minuet from the 6 suites, or the Breval sonata or something, like 6 months in?) go find a local luthier or bowmaker, see if they have a showroom where you could test-drive a $5k cello (that's middle intermediate, like high school players). they should give you a bow and some privacy. See how it feels compared to what you're used to. It's hard to gauge sound under the ear. If you're shopping for sound, you need to take someone with you (a teacher) or have someone at the shop play them for you while you sit across the room listening. But you can definitely feel the difference. They will probably come with Larsen A/D and Spirocore tungsten C/G strings, they'll give you a $1k wood bow to test it with, and the end-pin will be sharp :-)

If you want to argue the difference between a $5k cello and a $50k cello, it gets harder to quantify and you'd have have to do it in person, not in writing. But from where you are? A $1500 yamaha cello would be a step up, but you wouldn't understand what you're missing. I'd say $3500-5k and preferably a locally handmade one (no "brand," just the maker's name inside). It's not snobbery, there's something you'd just have to feel, but you might not be at a place yet to tell the nuance.

Depends on your strings and setup.

That said, Hill Light Rosin.