Hi there, and welcome to the forum! I'm from Latvia, too - where are you located?
Horse "nails" are generally called hooves in English - just for future reference, so that you have an easier time understanding what is everyone talking about. ;) Hooves should be trimmed ("cut") once every 4 weeks. Is he barefoot (bez pakaviem) or shod (apkalts)?
Leg wraps won't do any significant good in jumping, so you can chose to use leg protectors, but I, for example, don't use anything on lower heights, and my horse is ok.
I would advise you not to cut his mane, as it protects the horse from insects in summer and drains the excess water away during autumn rainfalls, and also helps with thermoregulation in winter.
Two geldings are more likely to have playfights more often, to establish dominance, and a gelding might chose to mount a mare. However, I haven't noticed much difference if it's about just two horses. I'd chose a gelding, but that's my own preference.
If you don't clip your horse, you won't need a blanket if he's in good health, however, he should be able to go outside for all the day during winter - being paddocked for such a little time is harmful for a horses' health! He needs movement, fresh air, unlimited hay and company.
He needs a well fitting saddle, fitted by a professional. If the saddle fits well, then a jumping or an all purpose ("universal") saddle is just what you prefer.
I recommend you to read this article about horse well-being and natural needs, hope it helps (English readers - sorry, this is in Latvian :) ) - http://zirgam.lv/2013/08/13/jusu-zirga-pamatvajadzibas/