I think the verb "seeking" has the "for", preposition part, already 'built in' to it. Whereas looking is not the same as seeking.... but you'd need to look FOR something to be sort of the same as 'seeking'.
I think 'letar efter' is a verb with particle mostly like the English 'look for', they go together. But mostly with concrete, material things, I suppose. 'Leta arbete' feels more "Uncountable", less concrete, Maybe that is why.
No. You can look for ett arbete or ett jobb, because both these can mean 'an employment', but ett yrke is not an employment, it's just a profession. And it doesn't make sense to leta efter that. Maybe you are trying to välja 'choose' yrke, but you wouldn't leta efter it.
Swedish 'leker' is used when children 'play', e.g. pretending to do their parents work. I suppose you confuse 'leker' with 'letar' = looking for. We can use 'letar arbete', as well as the verb 'söker' = search for, that in connection with 'arbete' has the same meaning.