Letter Boxes

Place the numbers 1-9 and the letters A-I into the 3x3 grid, such that the given requirements are met. For example, according to the first requirement of Puzzle 1 below, the "A" will be placed two boxes below the "1". The last one means "I is not in the bottom row." Each box will hold one number and one letter. 
Puzzle 1:

+---+---+---+
|   |   |   |
+---+---+---+
|   |   |   |
+---+---+---+
|   |   |   |
+---+---+---+

+---+  +---+          +---+
| 1 |  | B |          | C |
+---+  +---+          +---+
|   |  |   |          |   |
+---+  +---+---+  +---+---+
| A |  |   | 2 |  | 3 |   |
+---+  +---+---+  +---+---+

+---+---+---+  +---+---+---+
| 4 |   |   |  |   |   | E |
+---+---+---+  +---+---+---+
        | D |  | 5 |
        +---+  +---+

+---+---+  +---+---+  +---+  +---+
|   | 6 |  | G |   |  | 8 |  | I |
+---+---+  +---+---+  +---+  +---+
| F |          | 7 |  | H |  |   |
+---+          +---+  +---+  +---+

and: E and 9 are not in the same box.

    
Puzzle 2:

+---+---+---+
|   |   |   |
+---+---+---+
|   |   |   |
+---+---+---+
|   |   |   |
+---+---+---+

+---+---+  +---+---+  +---+---+
| A | 1 |  |   | E |  | 6 |   |
+---+---+  +---+---+  +---+---+
           | 5 |          | F |
+---+---+  +---+          +---+
|   | 2 |
+---+---+      +---+---+
| B |          |   | G |
+---+          +---+---+
               |   |
+---+---+---+  +---+   +---+---+
| C |   | 3 |  | 7 |   |   | 9 |
+---+---+---+  +---+   +---+---+

+---+---+---+  +---+---+---+
| D |   |   |  | 8 |   | H |
+---+---+---+  +---+---+---+
        | 4 |
        +---+

and: I and 4 are not in the same box.

Source: Original. Similar to earlier "Letter Lines" puzzle.

Solutions were received from Henry Bottomley, Mike Rattenbury, Stephane Higueret, Philippe Fondanaiche, Samantha Levin, Sudipta Das, Cornel Margine, and Richard Karis. Cornel sent good step-by-step solutions (below.) 
  1. ************* Puzzle 1 *************

It's easy to notice that : 
 +---+          +---+      +---+---+
 | B |          | C |      | B | C |
 +---+          +---+      +---+---+
 |   |          |   |  =>  |   |   |
 +---+---+  +---+---+      +---+---+
 |   | 2 |  | 3 |   |      | 3 | 2 |
 +---+---+  +---+---+      +---+---+

and

+---+---+---+  +---+---+---+    +---+---+---+
| 4 |   |   |  |   |   | E |    | 4 |   | E |
+---+---+---+  +---+---+---+ => +---+---+---+
        | D |  | 5 |            | 5 |   | D |
        +---+  +---+            +---+---+---+

Then we have two possible cases:

+---+---+---+
|B 4|C  |E  |
+---+---+---+
|  5|   |D  |
+---+---+---+
|  3|  2|   |
+---+---+---+

OR

+---+---+---+
|   |B  |C  |
+---+---+---+
|  4|   |E  |
+---+---+---+
|  5|  3|D 2|
+---+---+---+

Because E and 9 are not in the same box we search for other digits that can be with E,
then we search for digits that can be with the new added letter (added together with the last digit)
and so on... till we find or not a complete solution.
The first case doesn't give any solutions. But from the second we get:
+---+---+---+
|G 1|B 8|C 9|
+---+---+---+
|I 4|H 7|E 6|
+---+---+---+
|A 5|F 3|D 2|
+---+---+---+

  2. ************* Puzzle 2 *************

From 

+---+---+---+
| D |   |   |
+---+---+---+
        | 4 |
        +---+

and

+---+---+---+  
| C |   | 3 |  
+---+---+---+  

we have 2 possible cases:

+---+---+---+
|C  |   |  3|
+---+---+---+
|D  |   |   |
+---+---+---+
|   |   |  4|
+---+---+---+

OR

+---+---+---+
|D  |   |   |
+---+---+---+
|   |   |  4|
+---+---+---+
|C  |   |  3|
+---+---+---+

We do the same we did in the first puzzle, only this time we search for letters that can be placed in the
same box with already present digits. The solution is:

+---+---+---+
|D 8|G 9|H 2|
+---+---+---+
|A 6|B 1|E 4|
+---+---+---+
|C 7|F 5|I 3|
+---+---+---+
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