1) Погодить is informal and, as far as I can tell, rarely used nowadays except for imperative "Погоди" (=wait a bit). Подождать is also used for that, and for everything else as well.

2) You certainly missed a lot. There are many Ь-ending nouns in Russian. There are masculine amongst them as well as feminine ones. Techincally, if a word ends in a soft sign, it just ends in a soft consonant. If the word in masculine - OK, it just declines as any other consonant-ending masculine noun. If it is feminine, it has a separate pattern. If it is an abstract noun, it is most certainly feminine, like относительность, смелость, хитрость, мудрость, старость, слабость, боязнь, власть, суть. All abstract nouns ending in -ость/-есть are of this kind.
Of the more used masculine nouns I can tell гвоздь, писатель, учитель (-тель is an "-er" suffix), выключатель/включатель, словарь, день, дождь, фонарь, рубль.

All month names (Январь through Декабрь) are masculine, as well as all words with suffix -тель. Beware of feminine "обитель" (dwelling-place), though. A rare word, but -тель, obviously, isn't a suffix here.
All nouns in -жь, -шь, -чь, -щь are feminine. Actually, it is why they are spelt this way: as you can guess, Ж,Ш,Щ,Ч couldn't care less if there is Ь after them or not. They are unpaired, anyway. Also -знь nouns, as far as I remember (болезнь, казнь, боязнь and so on)