If I begin casting Summon Monster in the surprise round, and I win initiative, can I finish the spell at the beginning of round 1?
Summon Monster spells all have a casting time of 1 Round. Typically, this means that you stand still, start chanting, and complete the spell just as you come into the start of your next turn. You risk losing the spell if hit before your initiative comes around.
In a Surprise round, if you're allowed to act, you (with a few exceptions) have only 1 Standard Action (that can be taken as a Move action) to work with. At a glance, it would seem that you simply can't use the spell in a Surprise Round. Luckily that doesn't make sense, and there's a rule for that:
Start/Complete Full-Round Action
The “start full-round action” standard action lets you start undertaking a full-round action, which you can complete in the following round by using another standard action. You can’t use this action to start or complete a full attack, charge, run, or withdraw.
This allows you to use two consecutive Standard Actions, such as your Surprise Round action and your first Standard Action to complete the summoning. Assuming you're not interrupted before your first Initiative (which you wouldn't be in your example), the creature(s) would arrive in the "middle" of your first turn and act immediately.
Other uses include moving to a safer location before beginning the spell. You just wouldn't be able to use your Standard Action the following turn for anything else.