There are a number of things you said that make it clear you are still a developing player, or what would in relative terms be called a "beginner" by the "good" players. No problem, every single player in the world, no matter where their skills and experience are at now, spent time going through that period (and sometimes it lasted for a while!). I only bring that up because it has some influence on what you should or should not do with your first cue purchase, and honestly you are on the right track from what you posted (many aren't, so that's good news!).
First, you don't know what your preferences really are yet, and you will over time almost certainly develop very strong preferences about various things related to what makes a cue the perfect fit for you. Second, although you will start to develop some preferences fairly early one, they will likely change some over time as you get even more skilled and experienced and have had the opportunity to try out a variety of other cues over time. Make sure you do just that at every opportunity, ask to hit with other people's cues when you can because that is one way you will ultimately find out what you like best, but remember your preference can and likely still will change as you get better.
The first thing I would say, and you already concur, is at this point you absolutely don't want to spend a ton of money because of the fact that there is a decent chance your preference is going to change before too terribly long if you stick with it (and you don't even know if you will even stick with it for that matter), plus, depending on the cue, once you spend more than somewhere in the $150-300 ball park range you aren't getting much more quality or performance, you are just starting to pay more for the name and/or for the additional workmanship that goes into the looks. There are plenty of quality options in the $150-300 range if you don't have to have certain names or looks. Don't let somebody convince you that you have to spend substantially more to have a good quality stick, or that only a custom cue will do as it just isn't really true although you may or may not want to do so later for other reasons.
The second thing I would say, since you don't yet know your preferences, is to stick to the specs that the majority find preferable and still find to be within the "usable" limits even in cases where their preferences turn out to be slightly different.
First, go with a 19 ounce cue, give or take a quarter ounce at most. Most players end up preferring a weight right around there, and almost all end up preferring a weight within an ounce of there.
For shaft diameter, 12.75 mm is indeed about a perfect starting point, and would be the diameter most would probably end up close to if low deflection were not a consideration (low deflection requires cutting down the weight near the tip end of the cue, which is typically at least partially accomplished by reducing the shaft diameter, so with the low deflection rage also comes smaller shaft diameters but it wouldn't be most people's preference for diameter otherwise).
Speaking of low deflection shafts, they tend to be substantially more expensive, and I wouldn't worry about them right now until you start to develop stronger preferences and are exposed to more things over time as you try them. Pool was played at the highest levels it can be played at without them for many decades, and still is today. If you find later that you like them, and are going to stay in pool long enough to justify the expense, then feel free to explore them but it isn't necessary at this point.
Get a standard pro taper. It is by far the most popular taper, and even for those whose preferences end up changing a little from that, it isn't usually much.
For a tip, if you have a choice, I would start with a medium. Many end up ultimately preferring that hardness, plus it seems to reduce miscues for newer players than what some of the harder tips will result in. And since you don't know yet what you will like, might as well start with the lower cost ones while experimenting and rule those out first. Triangle tips (it is called a "hard" tip but it really is more like a medium tip) are cheap, good performing, and might have been the most used tip over the past few decades and would be a great one to start with and adjust from there if needed.
Start with a 58" cue. It is ideal for most people and by far the most popular. If you have abnormally long arms you might consider a 60" cue but even for most tall people 58" is most popular.
Most prefer a slightly forward weighted cue that balances at about 19" or so from the butt end if you are able to try them out that way before buying, but this probably isn't a massive consideration right now and most aren't usually too far off from that anyway.
Joints and ferrules won't matter that much right now and/or preferences vary wildly and you just don't know what yours are yet so I wouldn't be real worried about them.
Same with wraps, you just don't know your preference yet. As you have done a lot of playing with house cues, if you already know that you like the wrapless feel that would be a great choice for that reason. If you find wrapless not to be perfect, an Irish linen wrap is by far and away the most popular so I would suggest starting there.
Butt caps are mostly for looks although you definitely want your cue to have a rubber bumper on the butt end to prevent it from damage when it touches the floor.
As for brands, there are quite a few that offer good cues in your general price range and I can't think of many off the top of my head so it would be best to ask here as you consider some, but a few that I can think of that you can get good cues in that price range are Schmelke, McDermott, and Pechauer. Schmelke will even let you customize things to an extent in that price range (they also allow you to leave off their logo if you find that you don't prefer the looks of it).