Chains are NOT significantly different per MTB versus road and not any different from component level regarding design and geometry. A high end Dura Ace or Sram Red chain may have hollow pins and different alloy and be 5-100 grams lighter and thus cost more. The difference you are concerned with are only speeds. 10 speed chains are narrower than 9 speed chains and 9 speed chains are narrower than 8 speed chains. Just stick with the same speed and make sure it states Shimano compatible. KMC, Sram make compatible Shimano chains along with a couple others.
The RD-T610 is compatible with the 105 CS-5700. Cranks are not usually an issue. Its more of a chain-ring compatibility regarding cross-chaining a drive-train. Remember that Shimano compatibilities are based their expectation that you may be cross-chaining a drive-train as shown in the illustration below.
In the real world, no seasoned rider shifts this way ever.
The illustration above show a cross-chained drive-chain with the chain represented in RED. This is an example of improper shifting.
The Deore RD-T610SGS is a long cage derailleur and is perfectly capable for taking up plenty of chain no matter the front chainring unless maybe you go to something ridiculously small like the smallest ring from a triple crank or MTB drive-train. This scenario could possibly pose issues.
The above illustration shows the derailleur while cross-chained and how it moves to its furthermost positions.
Since you have one chainring, this is not an issue. All you need is a 10 speed chain and you will have to remove some links for a proper fit.