Not exactly. 'Ithim' is a 'synthetic form,' combining ('synthesizing') the verb and pronoun into a single word; in the case of 'ithim,' that 'm' at the end is the first-person singular pronoun, 'mé,' which becomes simply 'm' when synthesized with the verb. The 'analytic' form (the form that keeps the verb and pronoun separate) is 'itheann mé.' That analytic form is more common in some dialects, but I have read that the synthetic form is the 'official' standard.
Same goes for the first-person plural: the synthetic form is 'ithimid;' the analytic form is 'itheann muid.'
Yes. Irish, like English, has a distinction between 'Paul eats bread (habitually, generally, etc)' and 'Paul is eating bread'. Some linguists even theorize that the English progressive developed from contact with the Celtic languages (I disagree with that analysis, though, as similar developments are found in non-standard dialects of other Germanic languages, so it seems to be a natural development). To say the latter you would say Tá Pól ag ithe aráin, using the genitive (literally translates to 'Paul is at eating of bread')