しかもやたらと散歩好きで、真冬だというのに、こうして一日二回の散歩は欠かさずに要求してくる。

、、。

On top of that, he's weirdly fond of walks — even in the dead of winter he demands his twice-daily walks without fail.

Literal

Moreover, unusually walk-loving-[and-で], midwinter [copula-だ] [even-though-のに], like-this 1-day 2-times [gen-の] walk [topic-は] without-skipping demand-[come-てくる].

しかも 'moreover / on top of that' piles on another characteristic. やたらと 'unusually / excessively / to an unwarranted degree' is an intensifying adverb suggesting the behavior is irrational. 散歩好き 'walk-lover' uses the productive ~好き suffix. 真冬 'midwinter / the dead of winter' uses the intensifying prefix 真 'true / real' (also seen in 真逆 'exact opposite'). 欠かさずに 'without missing / without fail' uses the classical negative conjunctive ずに. 要求してくる 'comes demanding' uses ~てくる in the 'does X in relation to the speaker' sense.