Effect of mixing acids on solubility
All strong acids dissociate in water: $\ce{HX + H2O -&gt; H3O+ + X-}$.
Metals reacts with hydronium ions, not with a particular acid, e.g. like $\ce{M(s) + 2 H3O+ -&gt; M^2+ + H2(g) + H2O}$.
( With nitric acid, it is more complicated due its reduction to $\ce{NO}$ or $\ce{NO2}$, but it is not important now. )
Metallic cations and acid anions are not paired with a particular counter ion, until they start precipitating. They all form together a &quot;ionic soup&quot;.
As a rule of thumb, the guidance is the salt solubility. But in case of mixed anions and/or cations and non stoichiometric cation/anion ratios, more important is solubility product
$$K_\mathrm{sp}=[\ce{M^n+}]^{m} \cdot [\ce{X^m-}]^{n}$$
The salts that would reach their solubility product would start precipitate,
but then hard-to-control kinetic factors of crystallization initiation and propagation take part.
The excess of the counter ion decreases solubility of the salt, compared to salt solubility in water. It is similar effect as gravimetric precipitation of very little soluble salts, where excess of the counter ion causes much lower ion residual concentration, compared to being just in water.
Other ions may alternate salt solubility by side reactions  ( if applies ) with the considered ions, or by affecting activity coeeficients - see below.

For the case 1, sulphates would have the minor effect of decreasing somewhat the activity conefficients that therefore somewhat increased solubility of the phosphate. But it is general aspect of ions presence.
For the case 2: It depends on if there is formed a double salt like LiCdPO4 or not.

In particular specific cases, there is possibility of forming double salts, where precipitate consists of integrated more than 1 cation or anion. E.g. crystallization from solution containing $\ce{Fe^2+, NH4+, SO4^2-}$ may lead to crystallization of &quot;Mohr salt&quot; $\ce{(NH4)2Fe^{II}(SO4)2 \cdot 6 H2O}$. Similarly for alums with $\ce{K}$ and $\ce{Al/Cr/Fe^{III}}$ as  $\ce{M^{I}M^{III}(SO4)2 \cdot 12 H2O}$.

During rapid crystallization there occurs also coprecipitation:

There are three main mechanisms of coprecipitation: inclusion, occlusion, and adsorption. An inclusion occurs when the impurity occupies a lattice site in the crystal structure of the carrier, resulting in a crystallographic defect; this can happen when the ionic radius and charge of the impurity are similar to those of the carrier. An adsorbate is an impurity that is weakly bound (adsorbed) to the surface of the precipitate. An occlusion occurs when an adsorbed impurity gets physically trapped inside the crystal as it grows.


In the case of rather concentrated solutions of well soluble salts, there is complication we should consider the ion activities  instead:
$$K_\mathrm{sp}=a_{\ce{M^n+}}^{m} \cdot a_{\ce{X^m-}}^{n}$$
Acticity coefficients for diluted solutions can be calculated by Debye-Huckel theory, but for concentrated solutions it is hard to theoretically predict them. We can use concentrations for rough predictions, but the numbers may not fit well.