Finished goods - relax those and you open a Pandora's box. Its an open question whether a robust argument arriving at similar conclusions can be provided without that restriction, but in the past hundred plus years, there hasn't been. The H-O approach of neoclassical economics rests on counter-factual assumptions, so it only says what would be true if we were a different species of animal entirely living in an economy organized on different lines.
So for that, the answer is, I don't know. I don't know of anyone who has successfully relaxed it, but it may be that the ones who have tried have been using tools that only address fictitious economies, while those who use tools that address real world economies have not actually been trying.
Balanced trade - no, that can't be relaxed and ensure the Ricardian result. Without balanced trade, there is either unrequited transfers or credit/debt relationships. When there are unrequited transfers, that blows the "all participants gains from the increase in production possibilities" line of the argument, and when there are credit/debt relationships, that brings in intrinsic uncertainty about what one or the other side of the transaction is actually getting in return. Indeed, one of the main ways that neoclassicals and their heirs bullshit themselves into thinking that globalisation is covered by the Ricardian argument is by working with models in which credit/debt is simply arbitrage over time, ignoring that fact that completion of the transaction takes place in an uncertain future under unknown conditions.
Self-sufficiency - no, that can't be completely relaxed - that is the immiserating trade literature, it all hangs on trade dependency allowing one participant to be forced to accept outcomes because of the incapability of declining to trade at all leading to a fundamental inequality in economic power. However, as I noted, if rivalry between multiple sources of the things that the dependent participants are dependent on, it can be mitigated. Under post-Industrial Revolution political economies, that dependency has been on capacity to produce capital equipment, though of course the specific strategic capacity that defines core, semi-peripheral and peripheral economies in a trading system will vary over longer stretches of economic history. I've been accused of being a Marxist, yet while Harpo's my favourite, it's Groucho I'm always quoting. Odd, that.
