When submitting a word, you don't have to tap exactly on the green check mark. Tapping anywhere on the white bar that the green check mark is part of will submit the word.
In the Bear Selection screen, you can add a bear to your team by double-tapping its picture in the grid (you don't need to first tap it to select it, then expend the energy moving your finger over to tap the Select Bear button). Likewise, if a bear is already on your team, you can remove it by double-tapping its picture either in the grid or in the small circles where your selected team is shown--you don't have to click the Remove Bear button.
The Time Warps and Bear Selection screens have TV-shaped icons in the upper-right that will take you to TV Time if you tap them. These will save you a few taps if you realize you're about to enter a match or use a time warp but forgot to activate the benefits that TV Time provides.