is it courteous/acceptable to approach the smaller company and ask them to match the larger offer?

Above feels slippery.

To stay on the safer side, I would at least use different wording than "ask them to match". Say, informing them on higher counter-offer (purely informing - no pressure please) would be both courteous and acceptable, I would definitely do that. There could be further negotiation involved down this road, but it is rather impossible to give a one-size-fits-all advice here.

2. In the event that I accept the smaller company's offer, the large company asked me if I would consider contracting my services to them for 6-12 months after my departure...

First thing I'd consider in this case would be to consult with that smaller company. I would try to avoid risking my relations with the guys I am going to spend next few years working with.

A nice "side effect" of this would be an extra-proof that I can be a valuable asset indeed. This may be helpful since in the beginning of the career there is often a degree of uncertainty whether the hiring decision was right.

A word of caution: when negotiating in the bidding war, keep your eyes on mid-term perspectives, not only the short-term ones.

As an example, salary I've got thanks to bidding war put me into the grade level higher than I was really ready for. As a result, I spent two quite tough years trying to catch up to the required level before this discrepancy bites me.

And this is not specific to me or the company I landed in: many of my friends who happened to get to another side of "bidding fence", into another company, had similar problems too. They told me performance reviews turned out quite a hard game for them because of the non-favorable comparison to guys hired outside of "war period" being paid less for doing same job.

is it courteous/acceptable to approach the smaller company and ask them to match the larger offer?

Above feels slippery.

To stay on the safer side, I would at least use different wording than "ask them to match". Say, informing them on higher counter-offer (purely informing - no pressure please) would be both courteous and acceptable, I would definitely do that. There could be further negotiation involved down this road, but it is rather impossible to give a one-size-fits-all advice here.

2. In the event that I accept the smaller company's offer, the large company asked me if I would consider contracting my services to them for 6-12 months after my departure...

First thing I'd consider in this case would be to consult with that smaller company. I would try to avoid risking my relations with the guys I am going to spend next few years working with.

A nice "side effect" of this would be an extra-proof that I can be a valuable asset indeed. This may be helpful since in the beginning of the career there is often a degree of uncertainty whether the hiring decision was right.

_{A word of caution: when negotiating in the bidding war, keep your eyes on mid-term perspectives, not only the short-term ones.}

_{As an example, salary I once got thanks to some bidding war put me into the grade level higher than I was really ready for. As a result, I spent two quite tough years trying to catch up to the required level before this discrepancy bites me.}

_{And this is not specific to me or the company I landed in: many of my friends who happened to get to another side of "bidding fence", into another company, had similar problems too. They told me performance reviews turned out quite a hard game for them because of the non-favorable comparison to guys hired outside of war period being paid less for doing same job.}