I am considering getting a new Dungeons & Dragon set, and I am considering getting AD&D, but what really is AD&D? Does it involve more combat compared to 4e? Does it add more classes compared to 4e? Is it more roleplay oriented?
The traditional way of handling PC death in AD&D is for the player to roll up a new, 1st-level character. The bite of death is strong in AD&D, and the intention is that players treat the risks of adventuring very seriously.
However, what is traditional isn't universal—plenty of groups made up their own table rules for how to make a character after an advanced one died. You're well within your rights to say that they start with half their old PC's experience, or one level less than the lowest other party member, or at the average XP of the party minus 1000, or whatever simple or complex variation you can think up.
One of the things to keep in mind while considering how you want to handle this is that mixed-level parties not only work fine in AD&D, but they are expected. The power curve in AD&D is much flatter than in recent editions, making the difference between a 5th-level character and a 1st-level character much less than modern players might assume. With less power divergence, there is less mechanical pressure to ensure that levels are the same—so that should not be something you worry about when deciding how to handle new PCs joining an advanced party. Furthermore, because XP is split evenly among everyone who survives an adventure, any lower-level members of a party will advance slightly faster than normal, since their party will generally be taking on greater challenges—this makes the power difference even less of an issue than it already is.
Basically, history and I give you permission to handle this how you feel is right—there is almost no way you can handle it wrong. If you feel that the organic development of a character from 1st level is important, then you're in agreement with a lot of present and past AD&D gamers, and you should do that.
Try to ignore segments. They're a concept that was mostly introduced to keep track of how long it takes to cast a spell (more on that below), and aren't helpful for the rest of combat.
Instead, concentrate in the actual initiative results. Treat the action of a round as mostly simultaneous, but with the winners of the initiative getting the advantage. So it's not really that the losing side stands still for 30s, it's that they both maneuver, but the side with the initiative simply maneuvered better and got the battle line they wanted.
So in general, just use initiative to decide what order to resolve actions in. You can layer on house rules on this to make it work more how you think it should, but it's not necessary. (One house rule you'll see sometimes is resolving all combatants' movement in initiative order first, and then resolving actions in initiative order, to make the feeling of everyone just standing around go away.)
As for spells, a spellcaster already in mid-spell from an earlier round doesn't act on the initiative number of their side. Segments are counted to find out when the spell finishes, and until it does they are occupied. This often won't matter, as many spells are fast enough to begin and finish before the enemy acts or the round ends, but when the initiative rolls are close, then it matters, and when the spell takes a lot of segments to cast, it matters more. The most important thing segments tell you is: does the magic-user get hit before the spell finishes, disrupting it? and, does the spell take up their next round as well? Segments exist almost entirely as a tool to determine the answer to those two questions.