Can I draw more current from this battery?
The 'initial current' mentioned is not related to its load. It is the maximum permissible charging current when a charger is first connected.

35AH would be the C10 rating (continuous 3.5A for 10  hours). 

Going by the specifications for similar batteries, the C5 rating would be 30AH, C3 25AH, C2 24AH &amp; C1 20AH.

The number of charge/discharge cycles over the life span of a deep cycle battery would be 150-200 cycles at a depth of discharge of 100%, 400-500 cycles at 50% and 1000 or more cycles at 30%. 

Reference: https://batteryuniversity.com/learn/article/lead_based_batteries

This battery may be used with those loads staggered and with low duty cycles.