Mash's name is actually most likely derived from "Matthew"! which we can assume because in the first year's valentine's event she wrote her name as such on the cake she makes for it (it's fixed to "Mash" the year after)

AS FOR MASH AND LILITH (LiliMash? Mashlith? idk)

well first of all I should be up-front in saying: I don't speak Japanese and I can't read it either so everything I am about to say is based off of fragmented secondhand information which may not be perfectly accurate BUT you don't need much more when you have enough yuri madness-

The scene that covers the main thrust of their beef is here (starting at 6:33), but if you don't have time to watch it the tl;dr is Lilith asked Mash to figure out how she feels about her and Mash figures that she hates her, which she confirms and then goes on to say what she loves about humanity is that they're fickle and imperfect and emotional and hates that Mash's inability to bring herself to truly hate anyone goes against this, that it makes her too pure, so she hates her for it

"this is a lot of hate"

YES, HOWEVER- an important piece of context is that Mash is a homunculus, she was grown in a lab for the express purpose of working for the future of humanity as a demi-servant, that taken with Lilith being a demoness this means both of the women in the conversation have had experience in being treated as inhuman, so while the text is "I hate you for your purity" you can subtextually read it as "I hate you because you're wasting your chance to be human"

Now, building on this, we have Lilith's 3rd NP: "To You, A Misled and Faint Prayer"

While it's already fruity as hell to dedicate an entire NP to a specific individual ("A present for Mash Kyrielight" like, okay lesbotron5000), The part that makes me lose my mind is "I'm sure I will always hate you. Just like you will always hate me too"

If you just take this at face it's "we will always hate each other" BUT with the additional context that one of the things Lilith loves about humanity is their ability to hate one another and considers it a quintessential piece of the human experience recontextualizes it and allows it to instead be read as "I will always treat you as human, and you will always treat me as human in turn"

"so do they like each other or hate each other?"

Yes!

It's messy and it's complicated and they're so similar but so intractably different and they want to kick the shit out of each other and that conflict pushes them apart but also brings them so so close and

"a demonic Mesopotamian deity with the "head of a lion, the teeth of a donkey, naked breasts, a hairy body, hands stained (with blood?), long fingers and fingernails, and the feet of Anzû". She was believed to feed on the blood of human infants and was widely blamed as the cause of miscarriages and cot deaths."

so the head a teeth are a no, and her breasts aren't QUITE naked, but! she is quite furry, her hands are stained black with long fingers and nails, and while her feet are more mammalian she does still catch the bird-like features of an Anzu via her wings becoming avian!

also for comparison here's an engraving depicting an Anzu on the left (any direct depictions of Lamashtu are harder to find bc they got the Pathfinder deity in there a whole bunch when you look it up)

anyways I left in the bit of that description mentioning miscarriages and cot deaths, because, her bond CE, "Death, Death, Death"

the (unofficial) translation of the text in the description is:

On some day, at some time, in some place, a child died. And no, this event had nothing to do with you. The child simply died. Nobody needed to remember it, and nothing about it remained in history. Death, death, death. The world is constantly enveloped by it. Do you find it sad? Do you find it distressing? Sorry, but those feelings aren't real. The death of a stranger ultimately has nothing to do with you. In fact, it would be disastrous if you thought it did. If that was the case, humans would all too easily be crushed under the weight of the tragedy of death. But don't worry. I am the Witch of Night and Storm, and my memory is very good. So yes, I remember them. I remember the kind of life this child lived, I remember what made them happy, and I remember what made them sad. And so I will keep remembering all of the children that history has forgotten.Since I am the one who brought them their death, I have a duty to do so.

Now the Tough Part here is that Lilith and Lamashtu already thought to have some sort of historical connection due to them sharing a lot of similarities such as being evil Mesopatamian bird women and causing infant deaths, but also I can't find any description of Lilith in a cursory search that mentions fur or long stained fingers and nails at all, so, personally, I'd put my money on it being a little bit of both-