Seek for is not idiomatic English and is a contamination of to look for and to seek. You seek something or you seek to do something, but the preposition for is not normally used in combination with the verb seek. You could say:
- We are seeking people skilled in mathematics.
- We are looking for people skilled in mathematics.
Here is an example of seek + to infinitive:
We seek to expand our collection.
In your second sentence, the adjective different should follow life, because from this one modifies different and not life. Therefore, you should attach it to the adjective instead of splitting the postmodifier into a pre- and postmodifier by putting life in between:
A life different from this one.
There is a difference in usage for pre- and postmodifiers. Premodifiers precede the main noun of your phrase and are normally not too long. Having a long premodifier makes it more difficult to read the phrase. Often, long modifiers are put after the main noun as a postmodifier (although in scientific writing you can still find long premodifiers).
In your example different from this one is a rather long phrase, which is why you should use it as a postmodifier rather than a premodifier. If you only use the adjective different, you can use that as a premodifier. Simple adjectives are often put in the premodifier and longer phrases, such as relative clauses, are used as postmodifiers.
So, in your phrase with skilled, avoid a long premodifier:
We are seeking people skilled in mathematics.
In any case, do not split your adjective from other phrases that are modifying it:
We are seeking skilled people in mathematics.
In the above sentence, skilled still modifies people, but in mathematics no longer modifies skilled. As a result, you're saying that in mathematics is the location where you will be looking for skilled people.
Both of your sentences with long premodifiers sound odd. If you have no postmodifier, don't put in a long premodifier. You could perhaps make your premodifier slightly longer if your postmodifier is already terribly long, but in general the premodifier restricts itself often to just some adjectives, except for scientific writing as I mentioned earlier:
- We seek a much more pleasant life different from this one.
- We seek a much more pleasant life that is totally different from this one.
- We seek extraordinarily qualified people skilled in mathematics.
- We seek extraordinarily qualified people who are skilled in mathematics.
Note how the premodifier more or less restricts itself to adjectives (and adverbs) while relative clauses, gerunds and other phrases often go in the postmodifier. In all four of the sentences above, the premodifier could be moved and used as a postmodifier whereas it would not really work the other way around.
Additionally, it sounds rather odd to mention from this one before you mention life.
In short, don't make your premodifier too long, it will not improve readability. Restrict it to mostly adjectives and perhaps adverbials modifying those adjectives. Longer phrases should go in the postmodifier.