It seems to me for this game like there is a great deal of skill involved to have any hope of beating the sphinx or putting up a speedrun time for Hard (the only difficulty anyone has taken interest in thus far) but also, the luck of the gems that fall (even the same sequence for both players for fairness) no doubt contributes to the overall times possible in a given run.

There is one additional factor that also contributes to luck to me in this game: the behavior of the mummy and to an extent the sphinx, mostly with aggressive/conservative build strategies with the mummy.

Sometimes, for the mummy in particular, it will use the speed it has acquired by going through the pyramid to build high very confidently, raising the possibility of beating it very quickly with the right timing of attacks from people who would actually speedrun this on Hard, and quite a few victories under 30 seconds for one round have happened against this character. And I did once find out for sure the mummy was doing it this way both games while I beat them both. The mummy seems to be random with this build-high to clear an early chain strategy or a more conservative one more like the witch, making for more work for the player to build crush bars on this mummy and making it take longer.

Even the sphinx can sometimes get reckless like I described with the mummy, but it's really freaking fast, so one hesitation or mistake and my plan to defeat it could be ruined. Still, this could make for a miraculous comeback in the end, especially if you got a few "magic gems" which can be used to just crush bar the sphinx to death that way by just using them one after the another, or just one of two or two of three where you think that will be enough, especially if you have it coming up "for real!" I've used this strategy to defeat the sphinx, and did use one of those in my final game for my speedrun when I was at 10 lines clear (20 to get the magic gem normally, but occasionally sooner) and "carried" the resulting two crush bars on the sphinx to my magic gem "for real" and ultimately won after successfully using it. It is especially good combined with a barrier so the sphinx can't break it, particularly as it intentionally targets your magic gems, breaking it every time it gets the chance to!

So here's the information I thought could be useful for people looking at this.

So, what do you think are the overall roles of luck and skill are for this game as opposed to other games that are speedran?

Sphinx (this also applies to Mummy) has a tendency to build a chain, however knowing when that piece comes is key, bonus points if that color is droughted because the sphinx doesn't know what to do in that situation other than do a breakdown of the stack, however it's easy just to spike if you have another chain setup pre-emptively because the last 2 stack far right and far left quite often at the start which makes them extremely vulnerable for spiking, 60 points should be enough to spike them no problem, but 80 is optimal because they seem to get above 10 points before you can get the chance to lock them out.

I just got interested in speedruning this game and I wanted to know if you guys have any tips for the mummy and sphinx or just in general because what I've been doing so far is making single matches as fast as I can and I'm not very good at making combos

There is one guide from @SegaJunkie under "Guides" for this game that nicely explains how to set up chains on purpose. The part about setting up chains starts at about 7:48. And as you get more practice doing this, you can figure out more situations later on in games where you can set up chains like this.

https://www.speedrun.com/columns_iii/guide/b0lqm

Indeed, I taught myself to do so in order to beat the Mummy and the Sphinx. The main thing is to get used to intervening gems on what would otherwise be matches that can form chains, and this can even be sometimes used to turn mistakes into chains (or bigger ones!)

Using SegaJunkie's tactic, at the Sphinx, if the gems line up for you for an easy 3-chain, actually expect the Sphinx to do the same right with you! This is because of the identical sequences of gems (apart from magic gems, which follow different rules) that players get for fairness, so if it's that easy for you once you reach the Sphinx, it will be matched nearly every time. So what I do about this is, instead of using my 20-point attack immediately, I wait for the Sphinx to use its attack and for its flash gem to come up, then use my attack to throw the crush bars back at the Sphinx and break its flash gem. Then, the Sphinx often ends up cursed instead of me!

I have a speedrun time of 30:08 which is in 5th place, and I either go for a 3-chain at the beginning, or else more rapid-fire 2-chains depending on the gems. If the gems aren't right for a decently quick 3 (oftentimes with a drought of pieces with two of the same color), the Sphinx often doesn't know what to do about it, and can quickly pile its gems up trying to get that 3 instead of successive 2's like I would do with those gems, which can allow for a "quick kill" of the Sphinx that round! Especially so for a speedrun where item use is allowed, and I've got a Magic Gem item, or a few of them! Even better coupled with a Barrier item so that gem can't be broken!