Is it possible to buy UK train tickets for whole week?

I am going to get train every day from Gatwick Airport station to Horsham as well as return ways during the week.

The first question is whether you need to travel during peak times or not. If you do (for example if you are commuting to a course that runs with a standard working day) then a 7 day season ticket is likely your cheapest option. If you can stick to off peak times then buying an individual ticket each day may be cheaper on some routes (turns out on this particular one the season ticket is marginally cheaper even if travelling off peak).

I'm not sure what the peak time rules for that particular journey are offhand. Note that the peak time rules can be different for "off peak day returns" and "off peak returns".

Advance tickets may be cheaper still but they tie you in to a particular train and are limited in number, probablly not worth it for a journey like this.

May student card of non-UK country give me a discount? Or my age - 24?

There is the 16-25 railcard which is available to anyone in that age range. You should be able to buy it at any manned station with proof of age and a passport-style photo. Older full time students can also get it but the process for that is more awkward and likely to be impractical for someone studying outside the UK.

Unfortunately there are a number of caveats that mean it is unlikely to be of use to you.

1. Railcard discounts are not available on season tickets.
2. A £12 minimum fare applies for journeys starting between 4.30am-10am Monday to Friday excluding Advance Fares. This minimum fare does not apply on Public Holidays or throughout July and August.
3. The railcard itself costs £30. If you are only staying for a week and not making any long distance rail journeys it is unlikely to be cost-effective.

One option that will save you time but not money is to buy several (dated) tickets in one transaction. When I've got to buy for odd days (e.g. it's not worth getting a season ticket just before Christmas) I often buy that day and the next. I'm travelling at peak times; this may not be so easy if you're travelling off-peak. It also requires a staffed ticket office rather than a machine.