Is there a construction of surreal games that isn't related to actual games? Is there a way to construct games that isn't based on games? For example, construction of the surreal numbers. I haven't seen anything about Dedekind cuts that allow for the lower set to be larger than the upper set, so that approach seems like it won't work. I'm curious about this primarily because I'm not actually that interested in games like domineering or kayles & just want to be able to construct games (mathematically) independent of positions in specific games. Additionally, is it impossible to multiply games? Or has this just not been worked out yet?
Related: Conway games and Induction Principle for games [Note: I want to be able to handle transfinite & loopy games]
 A: 
I haven't seen anything about Dedekind cuts that allow for the lower set to be larger than the upper set, so that approach seems like it won't work.

The "Dekekind cut"-like construction of Surreal numbers is not actual Dedekind cuts, and if you want a construction of loopfree (non-loopy) games you can simply drop the inequality condition for the two sets.
Following the slightly informal treatment in Claus Tøndering's Surreal Numbers - An Introduction, a game is just a pair of sets of previously created games. This can be made more formal using ordinals, as in Definition VIII.1.1 of Siegel's Combinatorial Game Theory. Siegel basically defines the games with formal birthday (ordinal) $\alpha$ as $\tilde{\mathbb{G}}_{\alpha}=\left\{(\mathscr{G}^L,\mathscr{G}^R):\mathscr{G}^L,\mathscr{G}^R\subset\displaystyle{\bigcup_{\beta<\alpha}}\tilde{\mathbb{G}}_{\beta}\right\}$, and then a long game (a possibly transfinite loopfree game) is an element of any $\tilde{\mathbb{G}}_{\alpha}$.

Additionally, is it impossible to multiply games?

You can certainly apply the definition of multiplication of surreal numbers to arbitrary games in the above sense. The problem is that unlike with numbers (or "nimbers"/"impartial games"), the "product" of arbitrary games does not respect equality. If $G_1=G_2$ and $H_1=H_2$, then it is possible that $G_1H_1\ne G_2H_2$.

I want to be able to handle transfinite & loopy games

As celtschk mentioned in a comment, loopy games are defined elsewhere in Siegel, in Definition VI.1.2. Essentially, a loopy game is defined as $((V,E^L,E^R),x)$ where $V$ is a set (you could think of it as the set of positions of the game) $x\in V$ (the starting position), and $E^L,E^R$ are sets of ordered pairs of elements of $V$ (showing which position transitions Left and Right can do). For those familiar with graph theory, $(V,E^L)$ or $(V,E^R)$ are digraphs, and Siegel calls $(V,E^L,E^R)$ a "bigraph" and $x$ the "start vertex". 
