"Past participle" is unfortunate terminology, because it is ambiguous between meaning passive participle (after passive "be") and perfect participle (after auxiliary "have"). All the same, that's the traditional term. Usually these two participles have the same form in "-ed", or "-en", or maybe always, but neither expresses a past tense.
The "-ing" suffixes in your examples are from the conversion of a sentence into a noun phrase; nominalization, that is. For instance, "do you remember having known more than one" is from "do you remember [you knew more than one]", where the past tense of "knew" is converted to perfect "have" in a non-finite clause, and the "-ing" converts that "have" into a gerund.
It's a lie-lay conundrum. Transitive vs. intransitive.
Your first example is grammatically incorrect and should read:
Our focus lies on the electronic and mechanic sectors.
Your second example is correct.
A transitive verb takes a direct object.
In "I like fruits" and "she makes cakes," the verbs "like" and "makes" are transitive.
An intransitive verb does not take a direct object.
In "she spoke softly" and "I run fast," the verbs "spoke" and "run" are intransitive.
Lay is transitive, takes an object. For example: "She laid the notebook on the table."
lay
to place (someone or something) down gently in a flat position
Note the tenses of lay: “lay, laid, laid” (to place or put down)
Lie is intransitive, does not take an object. For example: "My dog lies here." "She lay unconscious on the bed." The past tense of lie is lay.
lie
to be located in a particular place
Tenses of lie: “lie, lay, lain” (to recline or remain)
There are many verbs which are both transitive and intransitive.
For example: "She has been singing all day." "She has been singing the same song all day."
Best Answer
Both sentences mean about the same thing, though I think they're using different senses of focus. In "My main focus is Spanish", the word focus refers to the thing that is being focused on; in "My main focus is on Spanish", the word focus refers to the relationship of the person to the thing that they are focusing on. Using the definitions from the Free Merriam-Webster Dictionary, "My main focus is Spanish" is using sense 5a, "a center of activity, attraction, or attention", whereas "My main focus is on Spanish" is using sense 7, "directed attention".
Either way, the sentence ends up with roughly the same meaning. However, depending on the context and exact meaning, I think one could be preferable to the other. For example, I think "My main focus is Spanish" tends to preclude "My main focus is subject–verb agreement", whereas a person whose field of interest is already known to be subject–verb agreement could well say "My main focus is on Spanish" to indicate that their main focus is on subject–verb agreement in Spanish. But this is a very minor difference.