How long does developer solution last within its bottle
Assuming the container is not made of glass and is air-tight plastic with negligible permeability to CO2, the developer solution will last significantly longer than, as of this writing, the average human lifespan.  In other words, it will keep indefinitely.  

However, I mean that as a literal answer to your question, "How long does developer solution last within its bottle?".  Put another way, you can imagine the time it spends sitting sealed in its container as not being counted for all intents and purposes.  The next time you open it up, whether it be tomorrow or in 20 years, it will be just as 'fresh'. It doesn't degrade in that state.

However, the time it spends exposed to the air DOES count, and will gradually cause it to go bad.  

The developer is nothing special, in fact, it's very simple: Assuming it is the kind that is 10:1 concentration, it's an 11% aqueous solution of sodium hydroxide, better known as lye.  And while the sodium hydroxide is very stable and will not degrade or otherwise vanish from the aqueous solution, your specific usage case is using it for only one thing: the ph.  The developer's job is to be a strong base and not much else.  

The problem with strong aqueous solutions of bases (this is not specific or unique to sodium hydroxide) is that they will pull carbon dioxide out of the air and dissolve it into the solution.  CO2, when dissolved in water (like, for example, in a soft drink) actually forms carbonic acid.  If you want something for it's high ph (like in your case), acids of all kinds are your nemesis.  It will neutralize some of the sodium hydroxide, and the more of it that gets dissolved into the solution, the lower the ph will get.  This will present itself as the developer becoming less effective over time.  How quickly it goes bad is not dependent on how long it sits on the shelf however, but simply how much time it spends exposed to open air where CO2 will slowly but surely dissolve into it.  

Fortunately, this problem has solutions.  Solutions to solution, you could say.  (Sorry, I just couldn't ...resist.  OK I'm done).  Assuming you have the concentrated developer, just mix up a batch of diluted developer, enough to use several times.  This way, you only expose the concentrated developer to the air briefly, when pouring some out to be mixed, while it is the 'buffer' container of dilute solution that you frequently open and close, but you'll use it all up before it gets too weak.  Now, you can make another batch from the original bottle, and will have only opened a second brief time.  Rinse and repeat.

But, just to be clear, it doesn't necessarily go 'bad', it just loses pH and stops working as well.  And sodium hydroxide is one of the cheapest chemicals you can buy, so this isn't like you're developing this using molten unobtanium or something.  

Whatever you do, don't fall into the trap of buying some pure lye/sodium hydroxide tablets or powder on amazon with the seemingly thrifty idea of making your own developer on dememand for almost nothing.  Pure lye reacts VERY strongly and generates TONS of heat when you add it to water, and unless you have the safety accoutrements of a real chemistry lab and survived a chemistry graduate program (which presumably means you know how to do this kind of thing without dying or disfiguring yourself), you should really not attempt this.  All that has to happen for you to suffer a serious chemical burn (which, by the way, hurt WAY more than a 'normal' heat burn) is some of the water splashing onto any part of you that is exposed.  And it will splash because of said reaction, and the parts that splash are probably going to be the super concentrated parts since it won't have had time to diffuse into the entire volume of water yet.

But, the real secret: The developer ideally is 0.56% NaOH solution, or anything else with similar ph.  Do you know what has similar ph and is much cheaper? Oxiclean.  1 oz in a gallon of water and you're good to go.  It contains sodium carbonate, which is actually what your sodium hydroxide developer forms as it absorbs CO2.  When you put Oxiclean in water, it will disassociate into sodium hydroxide and dissolved CO2 (carbonic acid).  Only, the ratio is very favorable and the ph will still be strong enough at that concentration.  And it is very safe to handle, to add to water, and even has bonus sodium percarbonate which will form some hydrogen peroxide, which will do nothing but good things for the developing process.  And it doesn't go bad in the dry form, so you can just whip up some developer as needed.  

Or just buy this, if you would rather do it properly. It's $12.  Compared to the other costs, nonetheless labor, involved in etching your own boards at home, the developer is really not worth even worrying about.  

As for the light source, the only advantage to using UV light is exposure time.  It will not produce better results or a sharper board/eatch/smaller feature size.  This isn't integrated circuit lithography.  It's cheaper, easier, and safer to just use a mundane fluorescent bulb.  The results will be identical, the only difference is UV will have shorter exposure times, and the difference is on the order of minutes.  Just get a lamp with a nice bright bulb, and after some experimentation you can determine the optimal exposure time for that lamp, and avoid all the headaches and health risks with using UV.  Plus, you would have to do the same with UV anyway (determining the right exposure time).  It will still be well under 20 minutes even without UV. 

One final note: I do not know personally, so take this for what it is, but a lot of people who seem quite good at home etching all seem to say that the best way to get good results is to not use sodium hydroxide based developer at all.  I recommend taking a look at this page.  Yes, Mike of Mike's Electric Stuff fame (check out his youtube channel, its good).  His advice is easily some of the best to be had on all things electrical.  He has some good suggestions on how to get great boards, and the most important one is not using NaOH developer.  Don't worry, he tells you what to use. I can't vouch for the advice itself, but I can say that Mike has never lead me astray on other electrical subjects, and if it were me, I'd put a good deal of trust in his PCB etching guide.

Good luck!