Disclaimer: this is an incomplete answer and may contain errors. Some of these procedures may also require the sun to actually set, i.e. may not work too close to a pole in the middle of summer. (This also assumes you aren't on a tidally locked planet at just the right distance so that the hemisphere of eternal sun is livable.)
How long a day on the planet is
Day length AFAIK can be determined over at most a few days using nothing more complicated than a crude sun dial and an accurate clock. The real catch is the accurate clock; because they are expected to have external time sources available, even a "high end" smart device may not have a very accurate clock.
The procedure is to set up your sun dial (which can be as simple as a relatively straight stick stuck in a relatively flat patch of dirt) and watch it for a day or several days to determine when the shortest shadow is cast. This gives you "noon", and also "north". Once you've marked this, just measure the time between noon and noon for several days to get an average. (or just once if you don't need much accuracy).
If your smart device has enough storage, and you can afford to do without it for some hours each day, you may be able to set it to take a photo every minute or so around noon and use the photos to determine noon. This could also give you a result in a little over a local day.
(Another option, as Mike noted, is to take pictures of the night sky with the device in an absolutely fixed position and find which two are most similar. The above is essentially the same procedure, but using the position of the sun for your comparisons.)
The planet's approximate angle of inclination respective to the ecliptic
Assuming the planet isn't wobbling or doing something similarly crazy (which might not preclude this method giving you an answer, it just might be wrong)... I think this may actually be quite easy, if you can either take or somehow fake a long duration photograph. Assuming the stars are fairly stationary relative to the planet's rotation, you just need to take a long enough exposure to get some good star trails (you may be able to fake this with many separate shots, but you might then need to manually track stars), then use that to determine the planet's axis of rotation relative to your current position. Then just take several readings of the sun to determine the ecliptic and compare notes.
Should be trivial if you can solve the previous problem. Ignore the sun, as its position is seasonally dependent. Instead, use your star trails to find a true (rotational) pole and compare the angle to that to the angle to the sun at noon.
Roughly what time of the year it is (e.g. predicting that winter is coming or spring is coming)
With the prior disclaimer about accurate timekeeping, if you are near an equinox, this should be fairly trivial to accomplish by making two marks on your sun dial (toward morning and evening, but measuring between two marks may be more accurate than trying to judge sunrise and sunset) and comparing the length of time it takes to go between these over a period of time. (Bonus: you don't necessarily need daily readings for this, but you do need readings separated by several planetary days; the more separated, the better.)
You can also compare your ecliptic to your poles to refine your guess. This can't give you the answer outright, however, as, unless you are near one of the solstices, on its own it will give two solutions. Both together, however, should give you a good guess, and may also be able to suggest the planet's orbital period (i.e. year). However, you may need to be able to do some calculus for that.
p.s. I really hope this "survival kit" is intended for use on sparsely populated worlds and includes a powerful distress beacon. (Or maybe it's only intended for use on something like a derelict ship?) Not because it will help, but because if the designers considered unexplored planets in their possible scenarios, they darned well ought to have included tools designed to figure this stuff out.