Here is some general background information. The relevant search phrases for this topic are generalized cardinal invariants or generalized cardinal characteristics, and the topic has a growing literature, emerging over many years. The topic has been studied as folklore for some time.

Here are a few specific resources:

• Brendle, Jörg, Cardinal invariants of the continuum and combinatorics on uncountable cardinals, Ann. Pure Appl. Logic 144, No. 1-3, 43-72 (2006). ZBL1112.03046.

• Cichon's Diagram for uncountable cardinals, Joerg Brendle, Andrew Brooke-Taylor, Sy-David Friedman, Diana Montoya arxiv:1611.08140.

• Brooke-Taylor, A.D.; Fischer, V.; Friedman, S.D.; Montoya, D.C., Cardinal characteristics at $\kappa$ in a small $\mathfrak{u}(\kappa)$ model, ZBL06643764.

• Slides for a nice talk by Luke Serafin, Cardinal invariants of the generalized continua.

What you call $\mathfrak{d}_{\omega_1,\omega_1}$ is known as $\mathfrak{d}_{\omega_1}$. Your concept of $\mathfrak{d}_{\omega_1,\omega}$ is less studied.

Here is some general background information. The relevant search phrases for this topic are generalized cardinal invariants or generalized cardinal characteristics, and the topic has a growing literature, emerging over many years. The topic has been studied as folklore for some time.

Here are a few specific resources:

• Dilip Raghavan, Saharon Shelah, Two results on cardinal invariants at uncountable cardinals, arxiv.org:1801.09369.

• Brendle, Jörg, Cardinal invariants of the continuum and combinatorics on uncountable cardinals, Ann. Pure Appl. Logic 144, No. 1-3, 43-72 (2006). ZBL1112.03046.

• Cichon's Diagram for uncountable cardinals, Joerg Brendle, Andrew Brooke-Taylor, Sy-David Friedman, Diana Montoya arxiv:1611.08140.

• Brooke-Taylor, A.D.; Fischer, V.; Friedman, S.D.; Montoya, D.C., Cardinal characteristics at $\kappa$ in a small $\mathfrak{u}(\kappa)$ model, ZBL06643764.

• Slides for a nice talk by Luke Serafin, Cardinal invariants of the generalized continua.

What you call $\mathfrak{d}_{\omega_1,\omega_1}$ is known as $\mathfrak{d}_{\omega_1}$. Your concept of $\mathfrak{d}_{\omega_1,\omega}$ is less studied.