What conditions would make SN1 products the major products over E1 products, and vice versa?
In a practical way, you would have to consider each particular case, there are tons of reasons that could make a reaction to yield mostly E1 or SN1 product (it will be always a mixture, that can vary from 50:50 to 0.01:99,9, I hope I'm making myself clear, they are always competing reactions).

But there are several factors that contribute to make one of the two more favorable than the other, for example:


The kind of the reactant employed: 

If you use hydrogen halides, ($\ce{HCl, HBr, HI}$) you will mostly obtain SN1 products.


If you use acids like $\ce{H2SO4, H3PO4~or~TsOH}$ you will mostly obtain E1.

Why is this? The reagents/acids of the first case, are very good nucleophiles. The reagents lisetd on the second case, are awful nucleophiles.
A good nucleophile is key for a substitution. But for elimination you only need something that can abstract a product in the alpha position to the leaving group.

On the other hand, very strong and small bases, can favour an elimination through E2 mechanism, with almost no substitution taking place. E1 mechanisms is usually observed only under mild basic conditions.

Solvents, like you say, also can favour one type or another:


In some cases, solvent acts a base for E1 reactions. (i.e. $\ce{EtOH -&gt; EtOH2+}$). Bulkiness of the solvent can cause both E1 and SN1 (in case that the solvent itself is the nucleophile, which usually happens) to be slower: it has to approach the compound that undergoes the reaction anyway. Actually is easier to approach a hydrogen than a carbon, so probably will favour SN1 over E1, BUT, do NOT take this as a determining factor at all. The alpha carbon from which you need to abstract the proton might be more or less hindered.


Solvents with low nucleophility also decreases the % of substitution reaction.

Both E1 and SN1 are favoured in protic solvents, in general.

To sum up, you will always have both products, but yes, there are factors that can be modified in order to maximize one of the two reactions. There are some general considerations, like the ones I posted, but in practice, each reaction is a world.