Get rows with most recent date for each different item
Skinning Q#1 independently and slightly differently than @ypercube 

with cte as (select row_number() over (partition by product_id,
                                       invoice_id 
                                 order by dt desc) as rn,
                    product_id,
                    invoice_id,
                    amount,dt
               from product ) 
select product_id, invoice_id,amount,dt
  from cte
 where rn=1
 order by product_id,invoice_id;

 product_id | invoice_id | amount |     dt     
------------+------------+--------+------------
 PROD1      | INV01      |      2 | 2018-01-05
 PROD1      | INV03      |      1 | 2018-01-05
 PROD2      | INV02      |      3 | 2018-01-08
 PROD2      | INV04      |      4 | 2018-01-08
(4 rows)


For Q#2, you are on the right track, but the SQL will have a cross join(gasp!) 

I think a function with a loop/cursor would be more optimized (i'll try that in my next free block of time)

--the cte will give us the real values
with cte as (select product_id, 
                    sum(amount) as amount, 
                    dt
               from product
              group by product_id,dt)
select p.product_id,  
       (select cte.amount --choose the amount
          from cte
         where cte.product_id = p.product_id
           and cte.dt &lt;= d.gdt -- for same day or earlier
         order by cte.dt desc
         limit 1) as finamt,
       d.gdt
from (select generate_series( (select min(dt)
                                 from product), --where clause if some products 
                                                --don't have an amount
                              (select max(dt)
                                 from product),
                              '1 day' 
                            )::date as gdt)  d
cross join --assuming each listed product has an amount on the min date
     (select distinct product_id
        from product) p
left join --since we need to fill the gaps
     cte on ( d.gdt = cte.dt 
             and p.product_id = cte.product_id)
order by d.gdt, p.product_id
;