This is something that you would have to negotiate with your GM. He might allow something like Spell Specialization feat for your summon monster ability giving +2 bonus to CL and thus two additional minutes to its duration.
I would strongly advise against multiclassing summoner as it would quickly weaken summon monster as its level would be much below yours.
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I just have to ask as to why you aren't using the Eidolon as a bodyguard at this point. It's really good at doing that, as a summoner bonuses come into play eventually for it being close by and it's easy to make it far more capable then anything that you can get with Summon Monster of the levels you can cast, and its not hard to make it both look and act like anything on that list of creatures for the most part. It's a summoned monster that's around all the time so I'm just really curious about why you don't want to use it?
Summoner and Alchemist don't really mesh together in any good fashion for a multiclass, they require different casting attributes and summoning focus does not function with lost caster levels.
The longer duration summons applies only to your summons from the SU ability of the summoner. the Alchemist stuff would not affect your Summoner summons at all (if that was the intent)
Really you need to use the Metamagic feat (extend?) to double the duration of your summons. But beyond the Summoner ability you wont ever get them lasting more than rounds per level.
Have you looked into the Broodmaster Summoner? This way you can use alot of smaller creatures that would always stay out to protect you (assuming your awake).
Having a summon monster spell be perm is a 20th level conjuration school ability so I dont think it would be appropriate for a low level character.
Meta-magic extend rod with the conjuration school would make you summons last 2 and a half rounds per level. For instance at level 4 you could have a small earth elemental for 10 rounds.
Other then that there is really only the summoner class that makes them last mins per level.